If you are just talking about keeping the house from falling down around your head, and a few minor adjustments or additions here and there, but no real renovations -- the tool kit is rather simple.
A pair of pliers.
A pair of clamping pliers.
A light hammer.
A general purpose utility knife.
A paint scraper and
a putty knife.
A pair of wire cutters.
Safety goggles, gloves and dust mask.
Two power tools should be part of this kit: a variable speed reversible drill (for drilling, sanding, driving screws?.) and a simple jig saw. The jig saw doesn't cut anything very well, but with the right blade it will cut absolutely any material from wood, to plastic, to metal, to ceramic tiles, to foam rubber.
After that you can grow, according to your likes and your necessities, into some specialty tools for electrical work or plumbing or woodworking. But wait until you really need them. Remember that large tools can always be rented.