The way to pray Salah has stayed the same since the Prophets time, with just minor variations depending on madhab.
I feel the best way to learn is to watch someone pray so you can see the movements correctly. This is one of the best videos I have found, demonstrating how to pray:
13 elements to each prayer
Each prayer constitutes of a different number of rakahs, but there are 13 elements that each prayer has to contain:
Reciting intention for prayer in the heart
E.g.: I intend to offer 2 raka’ah obligatory prayer of Fajr for Allah Almighty. It is done simultaneously with Takbeeratul-Ihram (below).
Saying الله أَكْبَر (God is The Greatest).
For the abled, leaning until it’s deemed not standing right invalidates prayer. If one is incapable of standing, he may sit, lay on his right, lay on his left, lay on his back or as he is able to do.
Reciting Surah Al-Fatiha
Recitation of Al-Fatiha is obligated for every raka’ah. Reading another surah after Al-Fatiha is also obligatory for the first 2 raka’ah for Farz namaz and in a Witr, Sunnat or Nafl Namaz in all the Rakats.
Ruku’ and stopping
Ruku’ is bowing the body until the palms is on the knees. Stopping means all major body parts including arms, wrists, head, legs stop – as long as saying “subhanallah”. If the body still moves, stopping is not done. An additional option is to read 3 times سبحان ربى العظيم و بحمده (Glory to my Lord, the Most Magnificent Most Praiseworthy).
I’tidal and stopping
I’tidal is standing again after ruku’. While the body is raising up, an additional option is to read سمع الله لمن حمده (Allah Listens to him who praises Him). During standing, an additional option is to read ربنا لك الحمد ملء السموات وملء الأرض وملء ما شئت من شئ بعد (Our Lord, to You is due all praise…). The body must stop as long as saying “subhanallah”.
Prostration involves putting these body parts – the bare forehead, 2 palms, 2 knees, the base of both feet’s fingers – on the place of prostration. The forehead must be bare, a covered forehead invalidates prayer. An additional option is to read 3 times سبحان ربى الأعلى و بحمده (Glory to my Lord, the Most High Most Praiseworthy). There are 2 prostrations, the second is done after sitting between 2 prostrations (as  below).
Sitting between 2 prostrations
During the sitting between the two prostrations, an additional option is to recite: ” Allah-hum maghfirli” Allah Forgive Me.
Reciting the final Tashahhud:
التحيات المبركات الصلوات الطيبات لله
السلام عليك أيها النبي ورحمة الله وبركاته
السلام علينا وعلى عباد الله الصالحين
أشهد أن لا إله إلا الله وأشهد أن محمدا رسول الله All greetings, blessings and good acts are from You, my Lord.
Greetings to you, O Prophet, and the mercy and blessings of Allah.
Peace be unto us, and unto the righteous servants of Allah.
I bear witness that there is none worthy of worship except Allah.
And I bear witness that Muhammad ﷺ is His servant and messenger.
Sitting for Final Tashahhud
The final Tashahhud must be done while sitting.
Greetings for Prophet Muhammad ﷺ (saws)
At least by saying اللهم صلى على محمد. At best اللهم صلى على محمد وعلى آل محمد كما صليت على إبراهيم وعلى آل إبراهيم وبارك على محمد وعلى آل محمد كما باركت على إبراهيم وعلى آل إبراهيم فى العالمين إنك حميد مجيد O Allah, bless our Muhammad ﷺ and the people of Muhammad ﷺ; As you have blessed Abraham and the people of Abraham. O Allah, be gracious unto Muhammad ﷺ and the people of Muhammad ﷺ; As you were gracious unto Abraham and the people of Abraham. Surely you are the Most Praiseworthy, the Most Glorious.
Greeting “peace be upon you” to the right side. At least with السلام عليكم, at best السلام عليكم ورحمة الله وبركاته
Following the Prayer Sequence
Prayer not in its rightful sequence is invalidated.
From  standing right to the  second prostration makes one raka’ah. These are repeated for every raka’ah. At the last raka’ah, the rest  to  is completed. (Note: For more than 2 raka’ah prayers, an additional option is to sit for Tashahhud on the second raka’ah.)
More Detailed Description Of How To Pray
This is a detailed description of how to pray a 2 rakat prayer, like Fajr.
Notes: When you are asked to ‘Say’ or ‘Recite’ or ‘Read’ in the directions. The sound must not be loud and at the same time not totally silent. It should be loud enough just for your ears to hear it and not so loud that a person next to you would be able to hear your recitation clearly. If a person next to you could hear only a mimic from you, that’s fine. If you are surrounded by noise such as a fan’s noise and you are not able to understand your recitation, it is okay. In an utmost quite room you should be able to understand your recitation. In any case must say it lound enough as mentioned above.
The reason we will be asking you to ‘wait a little’ through out the directions is to separate the Arabic qirat so that the meaning will not change. It is very important.
1) Stand up facing Kabah. It is called Qayam. The space between feet should be about four fingers apart, do not stand wide spread like most people do these days.. They should be pointing towards Kabah.
2) Say, “I pray two rakah (or as many required for that time) Fard (if its Fard or say Sunnat if its Sunnat or say Wajib if its Wajib or Say Nafil if its Nafil) for Allah Tala Face towards Kabah”. If praying behind Imam add “Behind this Imam”. It is called Niyat (Intention).
Note: It is Mustahib to say the Niyat. It is also permissible to make Niyat in your heart without saying it.
2) Raise both of your hands next to each ears. Touch the lobes of your each ear with thumbs.Then say “Allah Hoo Akbar”
3) Place the hands on navel right hand on top of left hand. Thumb and Pinky (smallest finger) should be wraped around the wrist of left hand’s wrist. It should be like you are locking or grabbing the left hand wrist. Rest of the three fingers of right hand should be strait in line to each other.
4) Now Read, “Subhana Kalla Humma Wa Bee Hamdika Wata Baara Kasmooka Wa Ta’ala Jaddoo Ka Wa La ilaha Ghairuka” Wait a little.
5) Then read, “Aaoo Zoo Billahee Minash Shaitaunir Rajeem” Wait a little.
6) Then read, “Bismillah Hir Rahmaanir Raheem” Wait a little.
7) Then read Soorah Fatiha, “Alhamdo Lillahi Rabbil Aalameen Ar Rahmaanir Raheem Maaliki Yaomid Deen iyya Kana Budoo Wa iyya kanastaeen ihdinas Siratual Mustaqeem Siratual Lazeena An Amtaa Alaihim Ghairil Maghdoobe Alaihim Walad Dualleen” Wait a little.
8) Then say, ” Ameen”. Wait a little.
9) Then read, “Bismillahir Rahamaanir Raheem”. Wait a little.
10) Then read any surrah from Quran or atleast three Ayats. We are writing you a small one. Read this Surrah, ” Wal Asre Innal Insaana Lafi Khusr illal Lazeena Aamanoo Wa Amailos Sualaihaatay Wa Tawa Sao Bil Haqqe Wa Tawa Sao Bis Sabr ” Wait a little.
11) Then Say “Allah Hoo Akbar” and bow (its called Ruku), hold your knees with your hands. The fingers should be separated with gaps in them. The back should be horizontally strait in line like a board.
12) Recite, “Subhaana Rabbiyal Azeem” At least three times. Three, five and seven times are Sunnat (Sunnat means what Prophet Mohammad peace be upon him liked to do). Wait a little.
13) Then start to stand up saying, “Samee Allahoo Layman Hamidah – Rabbana Lakal Hamd” Start to say Samee when leaving Rookh (bowing poster) and end with Hameedah when you are fully errect. Wait a little.
14) Now say, “Allah Hoo Akbar”
15) Now go to Sajdah (placing face on ground). In sajdah all the fingers of feet should be twisted so that the round soft part of the fingers is touching the ground. Hand fingers should be close to face and facing Kabah in strait line. Put enough pressure on nose so that the bone of nose should feel the hardness of earth (do not hurt your self but should be firm) Arms should be away from body like a bird opens the wings.
16) Now read, “Subhaana Rabbiyal Aala” at least three times. Three, five and seven times are Sunnat. Wait a little.
17) Now sit down on your left foot (It’s called Jalsa) the right foot should be sticking out and all of the foot fingers should be twisted. The soft part of the foot fingers should touch the ground. Sit for the amount of time you could say Subhan Allah at least three times.
18) Now go back to sajadah and perform as you did before Jalsa.
19) After completing second sajdah get up and go back to Qayam (standing up) Saying “Allah Hoo Akbar” Then read , “Bismillahir Rahamaanir Raheem” Wait a little
20) Now again read Soorah Fatiha, “Alhamdo Lillahi Rabbil Aalameen Ar Rahmaanir Raheem Maaliki Yaomid Deen iyya Kana Budo Wa iyyaka Nastaeen ihdinas Sirataul Mustaqeem Siratual Lazeena An Amtaa Alaihim Ghairil Maghdoobe Alaihim Walad Dualeen” Wait a little.
21) Then say, “Ameen” Wait a little.
22) Then read, “Bismillahir Rahamaanir Raheem” Wait a little.
23) Then read any surrah from Quran or at least three Ayahs. We are writing you another small one. Soorah Al kausar, “Innaa Aataina Kal Kauser Fasalli Li Rabbi Ka Vanhar Inna Shaaniaka Hooal Abtar”. Wait a little and repeat the steps from 11 to 18.
24) This time after second Sajdah do not go back to Qayam (standing up) but go back to Jalsa (sitting) but now it’s called Qaidah.
Now read, “Atthayyato Lillahe Wassalawato Wat Tayyebato Assalamu Alaika Ayyohan Nabiyo Wa Rahmatullahi Wa Barakatoh Assalamu Alaina Wa Ala ibadillahis Sualaiheen Ashadu An La ilaah illal Laho Wa Ashadu Anna Mohammadan Abdohoo Wa Rasooluhoo” Note: When you reach at Ashadu An La raise your index finger of your right hand upward facing about 45 degrees by closing the fist, and drop it back and spread the hand to its original position at illal Laho.
25) Now read this Durood Sharif, “Allah Humma Salle Ala Mohammadiyon Wa Ala Ale Mohammadin Kama Sallaita Aala ibraheema Wa Ala Aalay ibraheema inna Ka Hameedum Majeed. Allah Humma Baarik Aala Mohammadiyon Wa Ala Ale Mohammadin Kama Barakta Ala ibraheema Wa Ala Ale ibraheema inna Ka Hameedum Majeed.” Wait a little.
26) Now read this dua, “Allah Humma Rabbana Aatina Fid Dunya Hasanah Wa Fil Akhirate Hasanah Wa Qina Azaban Naar.” Wait a little.
27) Now turn your head towards right and say, “Assalamo Alaikum Wa Rehmatullah” Now turn your head towards left and read the same as above, “Assalamo Alaikum Wa Rehmatullah”
Salah is complete now.