Well, first of all, I am by no stretch of the imagination a fabulous bass player; but I think I do pretty well, and in the crowds where I run, I get a lot of postiive comments. But I guess if I've spouted off, it is reasonable to fill in a little background.
On double bass I play bluegrass, country, folk, old time (but not recently) and jazz.
When I decided to take up bass about 15 years ago, I followed the path I recommend to others. I rented a good quality hybrid bass and started taking lessons from a qualified instructor (MM from Manhattan School of Music). He asked me what my goals were, and I said that I was mostly interested in jazz bass, walking lines, etc., but that I expected to spend considerable time learning the fundamentals of bass according to standard methods. So we started out with a two-pronged attack: 1) Assiduous practice with the Nanny method and the bow; 2) Pizzicato/jazz training, formation of walking lines, transcriptions, and the like.
How I got into "roots" was that in the town where I lived there was a very active old-time scene, with a couple big big jams nearby. I started showing up with my bass. I knew that in the genre, simple was what was called for - and at the time, simple was what I could do. So I got started in the old-time community; but to be quite honest, I didn't find that music particularly compelling. However, from old-time it's just a hop skip and jump to bluegrass, and there's a big bluegrass scene in New England. From there it was word of mouth on "roots" styles.
At the same time, through word of mouth I got kind of accidentally hooked into the trad-jazz community in the area; again, simple is generally what's called for, and at that point simple was what I could do. So that was another venue for learning how to play good clear two-beat bass, with primary attention on rhythm and keeping the beat.
The thing is, that playing two-beat jazz bass, four-beat jazz bass, country, most types of folk, klezmer, bluegrass, old-time, and blues on the upright bass, all have pretty much the same requirements - strong sound, solid time, correct note choices, solid time, note duration, solid time, dynamics. And did I mention solid time?
When I see self-taught bassists in the bluegrass jams, they generally look like etiher guitarists or electric bass players, but sometimes they appear to have been attracted to the music and looking for the "easiest" instrument to get in the circle with. The stereotypical amateur self-taught bass player I see will exhibit a number of typical characteristics: - weak little strums perpendicular to the strings way up near their chest; - baseball bat grip and pooping out in just a few minutes if the changes call for anything other than open strings; - notes that kind of ooze out, at roughly the right time, but not a sense of actual timekeeping. I have been trying all these years to avoid all those. What I want to generate is a big strong sound, rock solid time feel., and to be able to play in any key. I'm not going to claim I've got that stuff nailed, far from it; but to the extent I've been successful, I attribute a lot of it to getting a clean standard foundation in the fundamentals from the get-go.
Now of course as with every instrument you can find people who play competently, or brilliantly, wtih non-standard technique. But OP is wanting to get started and (I assume) be playing as well as possible as quickly as possible. If your goal is to be playing in these "roots" genres, then you need to tell your instructor this is your goal; but in my humble opinion that doesn't preclude careful work with the bow to get the fundamentals ingrained.
One last bit - I am completely self-taught on saxophone; and now formally trained (at least the first couple years) on bass. I can tell you that when I decided to go my own way on saxophone, I ended up with bad habits that took me decades to get rid of; bad habits that probably would never even have gotten started had I afforded myself the benefits of formal instruction from the beginning.
But blues, country, and bluegrass aren't exactly rocket science... Don't get me wrong - bluegrass and oldtime are my favorite styles to play and listen to. But - certainly while starting off - they don't really require much in terms of unique instruction that you won't get from classical instruction. 
bbu.org
