Just got my first Amiga! A few questions...

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Here's what I would do if I were you:

Get a CV64/3D and scandoubler card.  That is, unless you plan to use the toaster...up to you  I have my toaster 4k in an a3k because I prefer to have the cv64/3d/scandoubler in my gfx slot.  I'm not a big fan of running the a4k through rgb...but that's just me I suppose.

The keyboard adapter that Cammy linked is not a bad idea, but you'd be better off scouring the interwebs for an a4k keyboard (IMO).

I'd also recommend cutting the battery out, scrub the area with some vinegar, let it sit overnight, and then you should be fine if there isn't too much damage.

One more thing to think of...I got ps2 mouse adapters for all of my miggies.  I absolutely loathe using a mouse that isn't optical.  I'm sure many folks can agree

Nice and easy upgrades for you would probably be an IDE to CF adapter, configure it inWinUAE and install it as your HD.  That's how I run all my miggies, and the a4k is probably one of the easiest to do that with.

Welcome!

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So just get the battery out with pliers? I don't want to have to deal with soldering on it until I have to, but that battery has to come out soon, hopefully it hasn't done much damage.

I use some strong wire cutters.  You may be able to use decent scissors, just be careful not to nick anything else  But yeah just snip the legs that connect it to the mobo and clean the area with vinegar or lemon juice.  Then wait until it's absolutely dry (like a day or two) before attempting to plug it in again!