salah prayer / tashahud

esselamu aleikum wa rahmatulah wa barakatu

i finnaly learned the full prayer now but now im not sure nomore if its the real sunni way!!!

so after second rakkat i use to say the first tashahud which is

At-Tahiy-yatu lil-lahi was-salawatu wat-tay yibatu, As-Salamy 'alika ay-yuhan-nabiy-yu wa rahma tullahi wa barakatu, As salamu 'alayna wa 'ala 'ibadil-la his-saliheen , Ash hadu al la ilaha illal lahu, wa ash hadu an-na Muhammadan 'abduhu wa rasuluh right?

and then after 4th rakkat i use to say the first tashahud again like above
and then i say socond tashahud which is

Allah humma sal-li 'ala Muhammadin wa 'ala aali Muhammadin, Kama sal-layta 'ala Ibraheema wa 'ala aali ibraheema innaka hameedum majeed, wa barik 'ala Muhammadin wa 'ala aali Muhammadin, Kama barakta 'ala Ibraheema wa 'ala aali ibraheema innaka hameedum majeed right?

but the problem is now that i read somwhere that this might be the shiite way to pray but i wanna pray the sunni way you know? so is this the real sunni way or not? and as opening dua i use to recite ayat al kursi is that halal?

and how do i do it in fajr prayer do i just say both tashahud after the second rakkat? i hope someone can help me!

Wa alaikum asalam wa rahmatu allah wa barakatuhu

You're praying the correct way inshaAllah brother, what you described seems correct.

During fajr, you just pray 2 raka's, saying both the first and second tashhahud at the end of the 2nd raka'a.

As for the opening dua'a, from my knowldege it is sunna, and not wajib, so if you get anything wrong it will not be counted sinful inshaAllah. I'm not sure however if one is allowed to recited quran during it. I usually say this:
"subhannaka allahuma wa bihamdiak, wa tabarak ismuka, wa taa'ala jadduka, wa la ilaha ghairuka".

Hope that helps, inshaAllah!

wajib means forbidden right? so its not forbidden and not sunna?
so its halal?

by the way ayat al kursi is the throneverse which is the most powerfull verse in the quran!

Wajib means obligatory. I don't think the opening dua'a is obligatory, but it is definitely sunna, and doing it earns you hasanat.

It is a great verse, yup.