Just use JQuery and have fun. Use it wiseley, use it cleverly!
Knowing every last IE6 browser quirk is not going to help in the long run, so just go for it man!
Whether it's a must-have nowadays is certainly more of a dilemma than it used to be for starting a new app but I personally continue to find it helpful for reducing DOM API boilerplate/cruft and a convenient way to get a UI team on the same page quickly. It's also an ideal model for any tool/library in that it stays out of your way when you don't want or need it to help you.
I personally still like it for the way it implements generic events, the fact that it's selector API can still be faster than the querySelector methods in a lot of cases (they appear to plug into the right to left evaluating CSS engines for most browsers) and its implicit iteration of collections which reduces a ton of cruft for common DOM-handling. I could write my own much slimmer version but it's already there and minimized to the size of a medium-sized jpeg so I don't.