Assalamualaikum wa rahmatullahi wa barakatuh,
The Sunnah of waking up is to recite the prescribed dhikr from the sahih hadith:
First: "Alhamdullilahil-lathee ah-yana bada maa amatanaa wa ilaykan nashoor."
Second: "Lah illaha illah lah wahdahu lah shareekahlah lahul mulk wa lahul hamd aw huwa-rabil-arshil atheem, Subhanllah walhamdullilah wa lah illaha illah wallahu ackbar wa la hawla wala quwatta illah billahi aliyul atheem rabbighfir lee."
The first is narrated in sahih bukhari and sahih Muslim.
The second one is narrated in Sahih bukhari and Ibn Majah. The reward for saying the 2nd one is that Allah will forgive your sins, and if you supplicate then your supplication will be answered, and if you perform wudu and prayer then they will be accepted.
These two are mentioned in the famous "The fotress of the Muslim" book compiled by a famous Islamic scholar: Said bin wahf al qatahni who recently passed away (inna lillah wa innah illaihi rajioon).
It is also mentioned that upon waking up one should recite quran verses 3:190-200, this is mentioned in sahih bukhari and sahih Muslim.
One last recitation that should be recited upon waking up is mentioned in sahih tirmidhi: "Alhamdu lillahil-lathee 'aafaanee fee jasadee, wa radda 'alayya roohee wa 'athina lee bithikrihi."
It is also mentioned in the sahih hadith that the Prophet (s) spoke of a man who woke up and said "Ya Allah I give part of my honor in charity to your slaves." (meaning he forgives those who wronged them, those he knows of and those he doesn't." The Prophet (s) recommended people to be like him, so to forgive upon waking up is a good deed.
After that it is confirmed sunnah to use the miswak to clean your teeth, then you can use the restroom and whatever, then perform wudu and pray 2 rakats nafil sunnah mokadoh (confirmed sunnah 2 rakat prayer) in your home and to keep it brief as Aisha (r.a) narrated in a saihh hadith that she saw the prophet (s) pray this prayer quickly that she thought he only recited al fatiha in it.
One opinion of the Islamic scholars is that you should recite only al fatiha in this prayer in both rakats. The other opinion is based on another sahih hadith where it narrates that the Prophet (s) recited surat al kafiroon in the first rakat, and then surat ikhlas in the second rakat (this is also the Sunnah for the 2 rakats sunnah mokadoh after maghrib prayer).
The reward for these 2 rakats is immense, some scholars even view these 2 rakats before fajr as mandatory. One hadith mentions that if you pray these 2 rakats in addition to the other 10 rakats of sonnah mokadh nafil prayer (4 before duhr, 2 after duhr, 2 after maghrib, and 2 after isha, along with the 2 rakats before fajr). Allah will build a house for the Muslim who does this every time they do it consistently, and the houses in Jannah are like castles. Another hadith mentions that the one who prays 2 rakats sunnah before fajr will get a reward that is better than the Earth and all that is in it. What that means according to the Islamic scholars is a difference of opinion. One opinion states that the Earth will wither away while these 2 rakats will remain. Another opinion is that if you took all the wealth on Earth and donated it in charity, then you would get that type of reward for praying these 2 rakats every time.
It is better to pray nafil at home (more rewared) as the Prophet (s) did.
The sunnah then is to go to the masjid (for men, women praying at home is more reward for them), and pray salatul fajr. The reward is that the man who does this will get the reward of Umrah (similar to it) based on one hadith, and in another hadith praying half the night (praying Isha in congregation gets the reward of hajj and praying half the night). The best prayer based on one hadith is fajr prayer in congregation on Friday.
The reward for fajr are many, specifically Allah will protect the one who prays it based on one hadith, and punish the one who skips it with hell fire (Allah threatens hell to those who skip it, the Muslims will go temporarily if Allah doesn't forgive them without punishment, the disbelievers will go forever).
Then there are many many prescribed Sunnah dhikr to recite in the morning, many of which are mentioned in that aforementioned book "Fotress of the Muslim" of which I do recommend Muslims here to memorize these dhikr, it shouldn't take more than 10 minutes to recite them which you can do in your car to work, you're allowed to recite the morning dhikr until 15 minutes before duhr.