RAW will give you the most potential to do the best processing. Whether or not you will be able to realise that potential is down to your processing skills and the software you use. It will not "automatically" give you better results. In the same way that shooting on manual will not "automatically" give you better images that shooting on auto.
If you shoot on RAW, you will need more cards, and your frame-per-second will be lower. Comensurably, if you are doing backups, you will need considerably more space, and a little bit more time.
Some photo hosting services like flickr will adequately backup everything you shoot in JPEG. For RAW you may need to find other means, e.g. burn CDs or mirror to another drive.
Generally, most people always shoot in "one way", e.g. just JPEG, just RAW+JPEG. It means you always have a familiar way of working. I wouldn't recommend trying to mix and match. You will inevitably end up cursing because you made the "wrong" decision for a particular shoot or shot.
In terms of how to deal with during processing, it sounds like you should do some real world tests. RAW isn't for everyone. You need to work out what it can do for you. So for this answer, you might have to discover for yourself...
...shoot some stuff with RAW + JPEG. Get some trial copies of the various packages that can process RAW. People here will have lots of good suggestions, or alternatively google a bit. Take the same picture and try your hand with the various packages. Ideally make it a technically tricky shot. Some areas of very fine detail. Some very shadowy bits, and some very light bits. And some vivid colours, and maybe fleshtones too.
If you can get a better (or at least comparable) result, with your processed RAW, vs. the JPEG, then you are beginning to find your answer.
An alternative view would be: Shoot it all in RAW anyway. If you don't have the skills or software now, you might get either or both in the future, if you ever want to come back to your "digital negs".
Personally I can't be arsed with RAW, but I came to that conclusion after I'd done lots of experiments to see if I could make it work for me. I'd strongly recommend giving it a try - your results, and your eyes, are arguably the final decider no matter what anyone on a forum says.