I have several basses, almost all have different strings on them. I do have 2 Precisions, one with Rotosound rounds, the other with Thomastik flats. Both sound fantastic in their own right. I've got another bass with GHS Pressurewounds, which are also great strings.
In terms of flats, I've only tried Chromes and TI's. I prefer TI's because of the lower tension. Chromes are a bit brighter. I haven't tried other brands of flats, but many of them are deeper than the two I've mentioned.
The GHS pressurewounds are just great. Very nice compromise between rounds and flats. I prefer them over ground wounds simply because I find ground wound strings a bit sticky. That said, grounds are closer to flats than pressure wounds.
Most nickel strings will sound less snappy than stainless steel, but still give you a "modern" sound without too much treble. I've used Rotosound, EBMM, DR, Fender, and GHS nickels. Honestly, they all are pretty similar. I tend to go to EBMM's because of price, availability, and plenty of string gauge choices to choose from.
After all that, I'd recommend you try the GHS Pressurewounds. If you find them too bright, you will know you want flats. Otherwise, they can cop pretty much most of the tones of rounds. If they are too bland for you, then you are a rounds player, my friend.