What to do after opening?

Especially with white, I often find myself in pretty good positions immediately after the opening, but I often just end up playing some random improving moves (Re1, Qd2, Rd1, etc.), maybe push a centre pawn or two to kick a knight, and within a few moves my opponent has equalised. 

Here's an example game where I get everything I want out of the opening, tried to start some attackey business, and it all fizzled out with one trade: https://lichess.org/MwMnawLD6EaR

Another good example game: https://lichess.org/hWDt2NpSh9mJ 

How do I keep up pressure in a good position, or more generally do better than random moves when in an equal position?

You need to understand the common themes of the opening, especially if you want to attack. Which are your strong pieces, which should you trade, what are the space advantages you have and what answers from the opponent cause weaknesses in their position. You need to understand the main ideas of your opening. May I ask, which openings do you play?

After the opening, the first thing I do is study the opening positionally. Who has a better position? If the position looks balanced then I look for any form of decent attacking opportunities. But you might be more suited to a more attacking style so if that's the case I'd recommend reversing my style.

In the 1st game you had an awfully strong center but didn't push, didn't take space and your opponent managed to trade your strong pieces.

I play the Ponziani, Alapin Sicilian and basically main line everything else for white. Normally I don't have the same issue with black, where I usually play the Caro-Kann.

What I'm getting so far is that I should have a big think after the opening though, yes?

I would say convert middlegames into endgames since people like me play better in endgames

In your first game, you built up a big center, but you really didn't use it offensively. When your opponent's pieces are developed in front of their pawns, you want to push them back with your pawns while gaining space. After 6... Qf6, you just developed your pieces and gave Black time to play ...d6, stopping your advance long enough to develop and castle. But if you had played 7. e5!, your opponent would have had to move the Queen, making it your turn again. (Clearly, 7. d5 is poor because it opens up lines for your opponent's Bishop and frees up e5 for his Knight.)
After you play e5, there's not really a good solution for Black. He brought his pieces out too early with no pawn cover, so he just loses time retreating. After the standard 7... Qd8 you can continue to push him back with d5, and all his pieces are cramped. After Qf5 or Qg6, you can develop Bd3 with tempo. After Qe7, where's his King Knight gonna go? Generally, if you can take space for free, you should do it unless you're afraid your opponent will counterstrike and break up your center.

After you missed 7. e5, Stockfish kept wanting to play a4, which isn't really a human move. Don't worry about it. Squeezing an attack out of a 1-point advantage in the middlegame can be really difficult for humans.

Make a plan! Target weakness, try to trade active pieces of the opponent. Try practicing and ANALYSE the chaotic games.

Ponziani and Alapin usually go for a big center (when possible), so try to cramp the opponent’s pieces: when you have the center (which often happens in the Ponziani and Alapin), your pieces have more mobility while theirs can be tied down. This spacial advantage can also lead to nice attacks, since your pieces can move to the right spots while the opponent’s don’t have as many options. When you have control of the center, try to protect your center pawns, as they can be useful for getting more space and possibly becoming passed pawns.

In your first game, you could have opened up the center, because your opponent was super underdeveloped (and their king was in the center).

good info

for middle-game plans, just when learning the opening, middle-game ideas apply for specific openings, like a kings indian for black, you need a pawn storm on the white king

Sometimes I try to open the centre after I castle but my opponent just castles too and I got rid of my centre for nothing: https://lichess.org/7kBr4jcm6ZWU

In this game there wasn’t much you could do to get a big advantage out of the opening. Sometimes we get great positions out of the opening, sometimes we get alright positions, and sometimes we get terrible positions (and all the in-between). Your position out of the opening was alright, but your opponent played well in the opening (like you did), so there wasn’t any way for you to get a crushing position.

Also, this doesn’t matter much (at all), but while the Ponziani is a good opening, it is not one of the very top lines. It is totally respectable, but with great play from both sides (White and Black), White has a tougher time finding an advantage than in certain other lines (like the Ruy Lopez or the Italian). While this doesn’t mean much at all (and I am not at all discouraging the Ponziani), sometimes there just won’t be much of an advantage to press for (as opposed to, for example, a mainline Ruy Lopez: that would usually give White a small opening edge). Only reason I say this is that if Black really knows what they’re doing… you might not get a big edge out of the opening (this game that you sent the link for, Black played well, so you couldn’t get much).

As for keeping the center, make sure it is worth it to give up the center (if you do that at all). Obviously, it is not always easy to know when it will work and when it won’t; over time though, your intuition will get better (especially for the openings you play), which will allow you to find when to keep it and when not to. If you’re keeping the center, just protect the pawns (as much as you can) and use the space advantage.