You should be able to get by with a 7-9 amph battery up to a 18 amph. You can decide to go with a SLA or Lithium battery for this. The larger you go in capacity, the longer you can go between charges. I run a 9 amph battery on my flasher, use it all weekend and really haven't one under 50% charge, so you should be good.
Your lights can be LED strips which are pretty easy to wire up. Most have an adhesive backing and you can also use zip ties to keep the strips attached to your poles.
You can decide to go bigger in battery size if you want to run more devices off the battery, but that's up to you.
If you are looking at power box options - check out Ice Hole Power - you can either purchase the unit made up, or they send you the parts and you assemble depending on your skill level - its really not hard to do. Also, lots of videos on YouTube (JDS Outdoors is a good option to review).