I don't know if there is a specific name for this structure or anything, but i've been curious lately about how to approach a problem I occasionally face.

In alot of games I play, no where in the middle game am I inclined to move one of these three pawns guarding my king forward.  I've noticed sometimes later in the game that this gives my opponent a solid target to attack.  Obviously when threat of mate arises I move a pawn.  I know that there are cases when that is simply the best approach, because during that turn, I am simply doing the move that best improves or least lessens my position.  My question is more about whether or not I should be making it a point to move one of these pawns in the middle game if it is possible later that my opponent will be able to threaten mate with it.  My question probably seems very broad, but my point is, how should I approach these wall, should I be looking for an opportunity to move a single pawn forward, or should I be trying to hold these pawns back when the position permits?  I realize that I likely won't get an answer that doesn't in some way say, "It depends" and that's fine if that's the answer, I just would like to get some thoughts on the subject.

Basically, as long as your king is over there behind them, you should avoid moving any of those pawns if at all possible. The strongest defensive set-up for you is just the way they are set up now (f2/g2/h2).

A few reasons for this come to mind: if you move your h-pawn say to h3, it can be a target for piece sacs (say with a N on f4 and Bc8) or pawn advances (...g5/g4). Both of these will open up your kingside to attack. The move g3 leaves large-sized holes for your opponent's piece to attack/settle on, and also weakens the long diagonal (a8-h1), often critically.  One problem with the moves f3 and g3 is that they can prevent your other pieces (particularly rooks and queens) from participating in defense along the third rank.  Another drawback to advancing those kingside pawns is that, the farther you move them up, the more room there is for Black's pieces to get in behind them. And one more point: the pawns are at their most flexible in their original position, so when (as sometimes happens) you do have to move one up to prevent some awful threat, you'll have a much wider (and better) choice of how to do this with them still on f2/g2/h2.

Thank you very much, that is extremely helpful!  I appreciate you taking the time and really giving me a basic/solid understanding.  As always I invite more on the topic, but I think that answered my question entirely.