So I am new to Islam. I took Shahada on the 30th last month. I have a question regarding prayers, mainly the daily prayers. Can I start off
with learning that and then branch out into other prayers?
Yes, of course. You don't have to burden yourself with more than what you can bear. Allah says in the Qur'an (what means), “Allah does not burden a soul beyond that it can bear… (Qur’an, 2:286)”.
I learned you gotta do Wudhu, etc and recite certain things and do
certain things before going to bed. I haven't yet completely learned
how to do each prayer, I'm still trying to write everything down!
Will Allah punish me for not doing them yet as I don't know them all?
First of all, let me clarify that there are only five fard (obligatory) prayers; two rakats of fajr, four rakats of zuhr, four rakats of asr, three rakats of maghrib and four rakats of isa. The duas (supplication) recited before going to bed are not fard which means skipping them is halal (permissible).
If you intentionally miss any of the fard prayers mentioned above, you will be held accountable for that. You don't need to know how to pray to perform prayers. You may/should write down all the instructions of prayer and read from that while praying.
P.S. This page has detailed instructions regarding how to pray salah.
Will I have to make up prayers (even though I haven't even counted how many days I may have missed it) or is there a prayer I can do to
repent and ask for forgiveness?
You don't have to make up the prayers you missed due to ignorance. But you should strive to do a lot of acts of worship and ask for Allah's forgiveness. Allah says in the Qur'an (interpretation of the meaning):
[39:53] Say, "O My servants who have transgressed against themselves
[by sinning], do not despair of the mercy of Allah . Indeed, Allah
forgives all sins. Indeed, it is He who is the Forgiving, the
[3.133] And hasten to forgiveness from your Lord; and a Garden, the
extensiveness of which is (as) the heavens and the earth, it is
prepared for those who guard (against evil).
And Allah knows best.
P.S. Recommended websites for new Muslims: