This is just my own $.02 worth, so please use my opinions only for what they are worth to you.
This is probably going to sound a bit crass or blunt, however I personally find that VR (regardless of which acronym you use) tends to be something of a crutch...if not comparable to a drug in that it tends to cause one to develop some degree of dependency, LOL! I don't own VR lenses myself...I've always been of the mind that as long as I don't have one, I simply won't miss it. For the difference one would spend on a VR lens, personally I'd rather rely on good technique instead...after all, a lot of great photographers did some really amazing work long before VR was around. Likewise, I would always recommend that a student or newbie learn that technique FIRST. In this case at least, learning how to properly hold and use a camera (and/or a tripod when necessary) will take one further than VR ever could. -IF- I had a lens with VR, I wouldn't thumb my nose at it, however, no...I won't go out of my way, let alone spend extra for one. I've been doing this stuff for quite a while and my hands are usually pretty steady (barring that second pot of coffee in the morning, LOL) and I can usually shoot pretty steady down to around 1/60 of a second or so (give or take focal length), so there's just not enough advantage for me to really warrant the additional cost. In all fairness, if the cost of the lens was the same, I'd probably opt for the VR with all other things being equal, but for the difference of a few hundred dollars (or more), I just don't usually see it as a useful investment.
Now on the issue of Sigma vs. Tamron, personally I'll go with Tamron every time. Sigma has made some very fine lenses over the years, however like Canon, they are also occasionally prone to Quality Control issues as well. Back when I was still a Canon shooter, I ended up with a couple of Sigma's that were basically little more than paper weights...the comparable Tamron's were FAR superior. I honestly don't know if that's still the case, however it really put me off Sigma. Like Mr. Photo up there, I also have a Tamron 70-300mm and it really is one of my "go to" lenses. I used this lens A LOT as a Canon shooter and it was the first lens I purchased when I switched over to Nikon. Likewise when I was still shooting Canon, I had a Tamron 70-200 f/2.8 (the NON VC version) and that was truly an outstanding lens in terms of image quality. No, it didn't focus as quickly as the comparable Canon L lenses at the time, however every single independent test I read said the Tamron was actually sharper than it's Canon counterpart...and I don't doubt that for a moment.
Also, if vibration reduction is in fact important to you, Tamron advertises its VC system as having a 3 coil system, which according to the hype is supposed to work better than other brands...if I'm not terribly mistaken, most of the competitors still only use a 2 coil system of some kind (I believe it's called "X and Y axis" compensation or something like that).
That said, on the Sigma vs. Tamron issue, my suggestion here would be; try both! Go into a local camera store or 2 and slap each lens on the front of your own camera...take a few pics around the store (even in the parking lot) at various focal lengths, then take the camera home and compare the images side by side...do a bit of pixel peeping to see which is really sharper.
BTW...for what it's worth, I do actually still shoot in Aperture Priority mode most of the time! I think that a lot of people have a misconception about this in that it's somehow "cheating" or that they are somehow less than professional if they use it. The fact of the matter is that I grew up with a manual SLR film camera. I still have my old Canon FTb...the ONLY thing the battery on the camera powered was a small on-board light meter. EVERYTHING was manual...manual focus, manual exposure and if you needed to change ISO or color balance, etc., you had to stop and change film (or carry an extra camera or two around with you)! These days, camera technology has gotten to the point that it's insanely good in the majority of average situations. I do -occasionally- shoot manual, if nothing else just to prove that I still can (or in the occasional tricky lighting situation...something that's heavily back lit for example), however the vast majority of the time I'd rather let the camera do the work so I can simply "focus" on the subject matter and composition (pun intended). My view is simply this; the camera (and all it's tech) is simply a tool. Like any advanced tool, it's worth taking advantage of. Think about it...if you were a carpenter and you had your choice between an old rusty hand saw or a state of the art, laser guided power saw, which would you use? You get my point. Perhaps this makes me something of a hypocrite in light of my comments about VR, but personally I just don't feel any need to shoot things in manual any more. As far as I'm concerned, AP mode is a REAL blessing!
Anyways, these are again my own opinions...I hope they help!