Ok, a few things here.
First off, the OP has a tuner, which is great! Count me among those who don't really count a tuner as an effect or a pedal. Anyway, he's got one, so time to move past suggesting that.
Secondly, the genre - stoner/drone. All the bands I've seen/known (which admittedly is not many, but some) playing that style tend to have some serious volume going on both at practice and on stage. This means that amp will not be cutting it. So, yes, thinking about grabbing a different amp is worthwhile. However, there is also a doomy sound quality to the bass in that genre, so I'd say there is a pretty good chance you'll be wanting/needing some dirt pedals sooner rather than later. There is a stoner/doom bassists thread on here that is very active and full of cool guys - pop in there, check it out, and ask those fellows questions about the genre that they play it, you should get some good answers.
Third...once upon a time when I was a wee little bass player, I bought my first dirt pedal before I upgraded an amp that I had, and I didn't regret it at all. The pedal sounded awesome, and using it was a massive step towards getting the tone that I wanted. So do whatever you want to. Yes, that amp is probably insufficient unless you intend to just be a bedroom player, so if you want to get out and start jamming with guys, more ampage is necessary. But if a pedal is where your heart is, go for that instead. Ultimately, you'll need both, so pick one to start with and the other will likely follow soon enough!
Fourth, the pedal.
@gregmon79 gave an awesome list of suggestions for dirt pedals back a ways, check that out and look into them. Also suggested was the TAFM, which is one of my favourite pedals ever. For stoner/drone, I'm going to say though that a muff type fuzz is probably your best bet. Don't get the EHX BBM, there are WAY better pedals you can grab than that one, for not much money (case in point - the Vick pedals). Look out though - once you get into dirt pedals, the rabbit hole is bottomless. So many different kinds and styles and flavours to choose from. You can own 100 different dirt pedals and still not have enough.

Oh, and as for the suggestions to get a compressor or something like that - pffft. Stoner/drone demands a dirt pedal first. Honestly, I wouldn't even bother with the multi-effects suggestions either, as in a lot of cases their dirt tones are not great. The caveat here though is if you are getting more ambient-droney and want to mess around with reverb and delay and stuff, in which case a multi-effect pedal could be cool.
Fifth and last (even though it should have been first), welcome to TalkBass!
