My advice is to buy tools as you need them. Eventually you will have what you need to get most any job done. The exception here is that you often want to buy certain tools as part of a set. For example, if you're starting from scratch, it would be silly to buy one screwdriver, or one wrench, or one drill bit. It's much more cost and time effective to get tools like these these in a set.
But like any DIY endeavour, however, you have to do your research. You can't just ask an internet forum which tools you'll need because we have even less of an idea than you do. Now if you were to ask, "oh hi guys, I want to build a bookshelf. Which tools do I need for this?" then we would be able to offer some specifics.
If you live near a Harbor Freight, they often have great deals. The only caveat I would mention is that you have to do your research here too. Some of their tools work every bit as well as the Snap-On version that costs 10 times as much, some aren't worth the paper their instruction manuals are printed on. Go to the website, read the reviews, inspect carefully before buying. Their hand tools tend to be of decent quality and most have a lifetime guarantee.
I would not even consider renting tools except for certain specialty tools that you'll use once every 5-10 years if you're lucky. (Thing something like a tile cutter.)