For the rules of pente, see GameHelpPente
Pro opening rules strategies
In the Pro opening rules (often known as tournament rules or simply Pente), the first stone of the first player must be placed in the centre of the board. The first stone of the second player may be placed anywhere on the board. The second stone of the first player must be placed at least three intersections away from the first stone.
For the 75 or so (disregarding mirrored positions) most likely opening positions where black's first stone is adjacent to the central stone, in only about 15% would a master level player be likely to choose to make a pair of adjacent stones on his/her third move (see Pro rules openings Fig. 1 for 10 example formations). In only two or three cases would a master level player be likely to make a pair if black hadn't already made a pair first. This is not to say white can't also win by creating a pair on the third move (either with or without the same second move) in other instances - white has 14 or 15 choices of third moves which form a pair after black's second move. Although some of them likely lose many are likely not explored much. A master level player would be more likely choose to form either an L or a hat shape (see bottom right corner of Pro rules openings Fig. 1) with his/her third stone, because these are commonly accepted white sure-win 3rd moves in these cases and there is little incentive for white to diverge from known winning lines unless possibly to avoid a potential novelty move by black in a live game.
If a pro rules opening is shifted so that an edge (or corner) of the board affects it, then the first player may well need to find an alternative third move to counter black's second stone.
Swap opening rules strategies
Opening proposals for Swap and Swap2 pente should consider the following principles:
1. White has the advantage of playing first and is therefore the first player capable of making five in a row.
2. Stones which are all located near the centre of the board generally favour white, because they allow room to expand in any direction. Stones placed strategically close to an edge or corner of the board can reduce white's first player advantage by limiting the directions in which white is able to build.
3. If white's stones are placed too far apart from each other and black's stone(s), black can often build potentials and shapes as though s(he) were white, because white will end up with at least one stranded stone which won't affect the game. As an extreme case, consider two white stones in opposite corners of the board with one black stone in the centre. White's stones are both useless for defending against black, and black is essentially the first player with two moves in the centre before white can defend.
4. Unlike in the Pro and Long Pro opening rules, white doesn't have the ability to choose the optimal placement of the second stone for handling black's first stone, so there is no hard and fast rule regarding how near or how far away from each other white's stones can be from each other before a position changes from a sure win for white to a sure win for black. There are definitely some sub-optimal second stone white placements three intersections away from the centre which can result in a first player loss, given an optimal placement of a black stone in the proposal. There may even be some second white stone placements which are closer than three intersections from the centre which don't create a pair with the first stone and yet are still first player losses. On the other hand, certain specific second white stone placements may prevail against certain black stones even though they are as far away from the centre as six intersections. There hasn't been sufficient study of all possible opening proposals with a white stone in the centre of the board to make a general statements such as more proposals with a Long Pro opening for white lose than win for white (although that is likely to be the case) or more proposals with a Pro (or closer) opening for white win than lose for white (also likely to be true).
5. If either player's stones are placed next to each other, they form a pair which can be threatened with a capture, which generally weakens the position for that player.
6. Diagonally aligned stones are generally weaker than horizontally and vertically aligned stones because there are fewer stone placements which create strong shapes available with a diagonal alignment.
7. Any opening proposal will favour one side or the other, so the goal is to find a reasonably balanced position which one is comfortable playing from either side of the board.
What can happen if white doesn't immediately play into the capture at c5 in the illustration below (Fig. 2) is well illustrated by the demonstration of how the pawnbroker shape can be used.
Differences in Strategies Between Gomoku and Pente
Due to overlines not winning in gomoku, there are certain board positions which would be a win for one player in gomoku, but the other player would win in Pente (see Fig. 4 for one example, h/t to pente.org). However, a far more common scenario which results in a position which is a win for one player in gomoku and the other player in Pente is the threat to capture a double keystone pair in combination with making a 3 (see Fig. 3). Ignoring the stone count on the board and just looking at the position after black's move to E6 in Fig. 3, we can see that in gomoku white would win the game, but in Pente black has the win. For an example of how a position like this can arise in a Pente game, see The Pawnbroker Boards illustrating these differences in who wins the game can also be found in this forum thread at pente.org
There are no draws in Pente unlike in gomoku, so a balanced opening proposal isn't possible for Pente.
Due to the potential for a pair to be captured (or threatened with capture) in Pente, it is frequently the case in Pente that forming an x_x with two stones is a better option than creating an xx pair of stones. However, there are some cases where the creation of a pair of stones is correct in Pente; in general, Pente openings for white where white's best third move involves creating a pair will lead the game towards pair exchanges until white arrives at a favourable stone position after the pair exchanges.
For board/game examples and more tips regarding the differences between Pente and gomoku, see also this forum thread at pente.org