How to get this vine thing off me

I think you're talking about the room with the red-and-white diamond barrel in Act 2 where you have to get the barrel low enough in order to progress to the next area. What I thought you needed to do was to jump up and down and time it properly in order to push the barrel down low enough.

This is the wrong method!

What you need to do is merely alternate pressing up and down on the directional pad in order to get the barrel to move, no jumping required. You can see a youtube video of this process here:

Do not feel alone! Pretty much everyone I talked to made this mistake as a young gamer. I never completed it either. I did get past the barrel twice I think, but by then I'd come so close to the 10 minute mark that I ran out of time before the end of the level. I didn't find out how to do this until talking with some friends a couple years ago. The answer eluded me for well over 10 years.

You can only do this if you are playing in Sonic & Tails mode. You need to at least have a second controller if you're playing by yourself. You need to make Tails fly by having him jump in the air, then hit the jump button again in mid-air. He'll then start flying and you need to keep hitting the jump button to make him ascend or release it to descend. After about 8 seconds on the ingame clock, he'll descend to the ground again because he's "tired." You can also kill enemies if you hit them with your tails.

While he's in the air, you can either have him fly over Sonic to pick him up or have Sonic jump up to grab his hands. Tails will move slower because of Sonic's weight. Note you can also do this underwater, as Tails can "swim" then, but you can't ascend while hanging onto Sonic.