Definitely get a solder sucker and braid, they're both good for different things. This is the solder sucker a lot of people in the diy pedal scene use. It's absolutely fantastic.
I only linked the UK site so you can see which one I mean and because it won't let me post a US Amazon link
I'd probably have a go at a few pedals first tbh just so you get some soldering practice in before having a go at your amps board. It can take quite a while to get good consistent joints.
Weller irons are good but also the Hakko fx888d gets a good rep.
Either use 60/40 leaded solder or lead free with 4% silver content. The silver lowers the melting point and makes it behave like leaded solder without the nasty lead part.
Oh and get some blu tac and masking tape. Both of these are great for holding things in place while you solder. Blu tak and the engineer solder sucker listed above were the two best tips I've gotten since I started in diy.