There is indeed much less expressed knowledge in Breakthrough than in Dots and Boxes or Hex, although my openings in Hex are almost clueless. There is already a thread about breakthrough strategy with valuable advice somewhere, but here are my 2cts for playing the opening and mid-game of breakthrough.
The two key concepts in breakthrough are zugzwangs and races. That is, most games are decided in a situation where one player wins a zugzwang, or a race, or both. Indeed, you can only get a piece advantage if your opponent blunders and forgets to capture.
(1) As some of you probably already know, the first consequence of the race concept being important is that you want exchanges to happen on your side of the board. If you assign an increasing value to the pawns as they get further from their starting position, you realize that you do not want to exchange a high-valued pawn (in which you have invested many turns) for a lower valued one (in which the opponent did not invest many turns). As a result, you usually do not want to move pawns pass the middle line in the opening of the game (my personal distinction between the opening and the middle game is when a pawn crosses the middle line of the board without being involved in a capture), because your opponent could force a disfavourable exchange.
(2) Note also, that attacking in Breakthough is much easier than defending as between 2 and 3 pawns are required to block an attacking pawn. You often want to leave pawns in the defending positions b1, c1, f1, and g1, because together they make it necessary for the opponent to break through with at least two pawns. You can also leave pawns on d1 and e1 (or b2, c2, f2 and g2) but this is less essential to me.
(3) So, what happens if both players follow roughly these advice. After some time, both player will have a couple of pawns on their third and fourth line, possibly on the second line also, as well as defending pawns on the home line. For at least one player, say White, it will become increasingly hard to make moves respecting the advice, because the opponent may have a good control of White's fourth line. If you wait a little more, White will be in zugzwang, that is, White won't have any move respecting the given advice and will either need to break the defending poosition, or to cross or exchange over the middle line, thus giving Black a significant advantage.
(4) My opening play revolves around this observation. I want my opponent to be the player in zugzwang. As a consequence, I strive for a good control of his 4th row (my 5th). Ideally, I'll get 4 adjacent pawns (or three on the edge of the board) on the fourth line before he does, for instance, imagine I have pawns on c4, d4, e4, and f4 and my opponent has pawns on c5, d5 and f5. Assume that these are well backed up, that is, we have pawns on b3 to g3 and b6 to g6. It is now impossible for my opponent to bring a pawn to e5 without an echange on my side or a sacrifice. The pawn on e6 cannot move because it is defending d5 and the pawns on d6 and f6 could move but it is unsafe as they would be defended only once when they reach e5. In this situation, I feel like I have won the e5 cell and I like my position, it will probably also be easier to win the b5 or g5 cells and my opponent will reach a zugzwang before me because they will have less space to make moves in.
(5) Now, all of this does not mean that you should always avoid bringing a pawn to the 6th row early. If you can develop an outpost on the 6th row (especially d6 or e6), it usually is good to do so as, because if the opponent wants to get rid of this pawn they will have to forget about their defending line.
(6) If you're going for the 4pawns defense (b1, c1, f1, and g1) but you're somehow disrupted and need to move b1 and c1 to b2 and c2, there now is breach through e3 and d2 which means that will also need to move f1 and g1 to f2 and g2. I value such a progression about 5 points (5 tempi), because you lose 4 tempi moving the defending line and the opponent second attacking pawn will not need to move as far as before (it will stop one cell closer to its home row).
Advice (4) and (6) are more geared towards advanced players, while advice (1), (2), (3), and (5) should be easier to apply by any player. There is also a lot to be said about the ending and computing races, but since you asked mainly about opening and middle game, I leave for another time.
I'd be really happy to see comments by other strong players. Even if it is just to confirm my perception or to inform me that they're not doing it like this at all!
Little Golem