As several people have noted, there are pretty good bass multi-effect pedals out there so if that's your cup of tea, go for it. As you can see, the Zoom gets a lot of love here.
If you're more into separates, then yes, you're right that a tuner is the #1 pedal. I like the TC Polytune 3 but there are lots of other options there.
How important EQ is depends on what bass you're playing—since you list that, I assume you're playing a passive bass without on-board EQ. You can probably get by with the EQ on your amp, but if you insist on a pedal, there are lots of options. There are about a zillion simple preamp pedals with standard 3-band EQ. If you need more than that, for a nice solid mid-priced EQ pedal I'd go Boss GEB-7. If you want a really fancy EQ, check out Source Audio EQ2.
Compressors are one of the deeper rabbit holes out there. Someone recommended the TC SpectraComp which is a good entry-level compressor. The next step up I'd look at the Keely Bassist and the MXR M87. Next step up after that you're looking at something like a Darkglass Hyper Luminal or a Cali76.
For a lot of genres, you will want some mild dirt, so an overdrive. If you just want a little tubey warmth but not much grit and you want to keep it on the cheap, you can't beat the TC Mojomojo. Spend a little more and you can get the TC Spark or the EHX Bass Soul Food or the MXR M89. And if you want to get up around $200 new, you're looking at things like the Darkglass Vintage Microtubes.
Octave pedals are reasonably common, at least in some genres. The new Boss OC-5 is supposed to be pretty good, lots of people like the EBS OctaBass and the various EHX POGs, but you'll get a lot of other recommendations if you want more there.
What else you need/want will depend a whole lot on what kind of music you play. For instance, if you're playing a lot of 80s, you'll almost certainly need a chorus. The EHX Bass Clone is a cheap chorus that sounds good, with the drawback that it has a pretty high noise floor. But for lots of genres you won't want a chorus at all.
As another "for instance," some genres (punk, metal, etc.) want pretty heavy dirt. However, there are so many distortion and/or fuzz pedals out there that sound wildly different it'd be hard to start listing without more information about what you're going after.
Depending on what you're doing, you might want a delay or an envelope filter or a synth or a reverb or a phaser or a looper or a... you get the idea.
Or, you know, skip all this and just stick with a single multi-effects pedal.