Buying tools as needed definitely works (I've done it that way, and probably get at least 1-2 new tools every 6 months or so for certain needs).
However, it is more expensive.
If you have a starting budget, it might make sense to get a basic bike mechanic's tool kit (the park one is excellent) then supplement with "as needed" tools as time goes on.
Never get rid of tools - I still have some of the tools I bought in the 70's, and am glad to have them (Sugino Cup/cone BB tools, for example).

There are definitely tools that warrant spending some money on.
- Cable cutters
- Allen (hex) wrenches - get the "T" handle style.
- Screwdriver - I've switched over to a style with a replaceable tip and use DeWalt or Milwaukee screwdriver bits.
- Chain tool - some may disagree, but a good quality chain tool makes life easier

Some other handy things to have in the workshop:
- handlebar stabilizer - simple but effective. This keeps the bars from swinging into the top tube. The cheaper solution is to use a long bungee cord looped behind the seatpost with the hooks on the bars.
- oil and grease. It is hard to go wrong with the Phil Wood grease and tenacious oil. If you have those two things, you are pretty much set for lube needs. You may have a preferred chain lube that is less tacky than the oil.

Once you are set up and able to wrench on your own bikes, and start to gain confidence in your capacity to do so (you'll find it is really easy once you get going), you will quickly pay for the tools in savings from LBS labor costs. Part of the joy of maintaining your own bicycle is a more intimate working knowledge of your bike, and a certain confidence that you can take care of whatever problems might arise.

Good luck!