Now that I think about it, I think I might suggest round wound strings. Flats can sound "dead", without much treble response (though the Chromes are brighter than a lot of others) - sounds great with a band for certain types of music, but if you're going to be messing around on your own then you may want to be able to have the full range of sounds that rounds provide - turn down the treble and you can approximate the flatwound sound.
Not to muddy the waters here, but I would have to respectfully disagree with you on this point. Not that I would recommend against roundwounds. These days it's probably safe to say that they are the standard string choice for manufacturers and most players and there is absolutely nothing wrong with them. However, I think it's a common misconception and oversimplification to say that flats sound essentially like roundwounds with the tone rolled off. I would argue that they are a totally different animal. Sure they have a generally darker tone, but there is also a lot to the flats sound that you simply will not get from a roundwound string. To my ears, rounds with the tone rolled off still lack the body and thumpy punch that is inherent to a nice set of broken-in flats.
At risk of getting super wordy, super early in the morning, regarding something that I've probably overthought for years now...
As a thought experiment of the physical properties of strings, imagine a calibrated set of tubes (or pipes -- not vacuum tubes), with one for each possible string gauge so that the given string could slide in and fit perfectly inside of the tube. As the name suggests, the outer winding of a flatwound string is flat, so when passed into the corresponding tube, the string would fill it completely with its mass. The outer winding on a roundwound on the other hand, exhibits a grooved texture. When fed into the tube of the corresponding gauge, these grooves would result in a whole lot of air inside the gauged tube, that would otherwise be solid string mass with a flatwound of the same gauge. So when thinking of strings of the same gauge, there is simply more mass to a flatwound string, not to mention the outer winding resulting in a certain stiffness which also causes the string to vibrate differently.
So I guess what I'm really trying to say here is that each type of string has very distinct tonal properties and that all players should take a stab at both styles. I'd also say that it's important to not just slap on a set for a day, but rather take some time with them, let them break in and play with the EQ on the amp. Settings that sound good for one set of strings won't necessarily sound good for another.
I know that when I first tried flatwounds back in high school, it blew my mind and I immediately started enjoying bass even more than I already did, and as a result felt inspired to play and practice at least double the amount I did previously.
I realize I probably sound like I'm here to preach the gospel of the flats, but realistically I'm just trying to stress the importance of exploring different types of strings. There's so much variation in tone and feel to be discovered there. Flats work for me, but rounds may work better for you.
Afterthought: I have a soft spot for Thunderbird basses and have owned a few (both Gibson and Epiphone). The interesting thing with these basses is that they are the only ones I've played on which I prefer roundwounds. TB pickups have a big mid-heavy, growly, though dark tone. I feel like they just shine with roundwounds and sound kind of dull with flats. I always went with a nice set of nickel rounds for their relative warmth in comparison to stainless steel, in addition to them not wearing on the frets quite as much, as already stated above.