DIY Basement Remodel
I can probably give you some general advice on basements from step 1 to 1000 but let's stick to the water issue.  

So you have a hill slamming into your house, you get water consistently during heavy spring rains but very minimal in the basement, and there has been little to no significant flooding over the lifetime of the house.  

In my opinion you can do all or some of these things:


Regrade your yard.  The hill coming down is fine.  But you should have a good 6-7 feet near your house that is running away.  You will need to grade your house to a notch on that side.  I had to do this to my house and I went from having a pond in my yard a few times a year and minor water in basement, to nothing at all.  I did stack french drains in the notched area (so I dug a good 6 feet down there).  But again this is very cheap to do and the most important thing.  Keep water away from your house.
Dig a perimeter drain system (there are a lot of products that can help you do this) and have it feed to a sump pump.  This is still very much DIY (1 day with a jackhammer) and cost is relatively cheap.  However anytime you start tearing up your floor that has had no issues, you have more than 0% chance of having issues.  Given the details you have provided I might dig a sump pump at the most offending location but I don't think I would dig out the perimeter floor.
You can waterproof that side of your house.  This is certainly very costly.  Not a lot of downside to this other than it usually will set you back maybe 2-3k plus for just one side of your house.  Again I think you have a grading issue and if that is fixed I think waterproofing may be overkill.


I would not suggest any sort of interior waterproofing including paints.  Let your concrete breathe.  Also I would suggest your framing is a good 1-2" from concrete and that you only insulate the top part of the walls.  If there is minor moisture, giving it airflow will keep it from becoming an issue.  The other thing is go with flooring that is either waterproof (tile or pure vinyl click-lock) or carpet that you don't mind possibly replacing (if you know that water is there carpet is easy to roll out of the way).