I was given a kit similar to
this one. This is a bit more expensive than you said you wanted to spend, but these kits are a good place to start and will allow you to assemble a bike. I've been adding tools along the way and never feel bad spending a bit of money for good tools.
Cable cutters are a good example. I had been using needle nose pliers with wire cutters to deal with cables, but recently got the Park Tool cable cutter. Big improvement.
Finally, for the love of Mike, don't use a screwdriver to change tubes. Dent your rims
and rip your tubes. Now THAT's a multitool.