What is some basic advice for playing blitz or rapid?

I can understand your situation. Because I am also a classical player, rated 2109 on lichess; but performing poorly on the blitz (1712) and bullet (1741). Moreover, there are more than 30 games, where I lost on time though the positions were in my favour. So, I think, I can give you some tips to improve your ratings by at least +300 points.

1. Smooth Transition: As per other answers, I think you should try 15+0 [Rapid] >> 10+5 >> 10+0 >> 5+5 / 5+3 [Blitz] >> 5+0 >> 3+2 >> 3+0 >> 2+1[Bullet] >> 1+0, in following manner: If you can complete (win/lose) 10 games in a row within the given time then go for the next tier.

2. One Opening man: Instead of experimenting with different openings, you must stick with a single one and simultaneously have to study it deeply. After studying a line properly, set a target to play first 10 moves within 10 sec. If you follow a single line (eg., 1.d4) for more than 100 games; then you will find that it is quite easy for you to reply not only fast but also accurate. And it will help you significantly in time management.

3. Pre moving: After "mastering" an opening, try to pre-move [This is crucial in the bullet, not in blitz]. Just make sure to play first 10 moves within 5 seconds!! and you will rock on bullet [at least below 1500 tier].

4. Pattern Recognition in Middlegame: A simple way to play ultrafast middlegame is pattern recognition (eg., if both King and Queen are in the same diagonal, then try to find your bishop to set a "Royal Fork"). The only way to achieve this is by practising a lot of puzzles. So, be prepared to try at least 20 puzzles (If possible 50) each day.

5. Accurate Endgames: "All's well that ends well." So, be focused at your endgame. First, master the checkmate patterns (Staircase mate, smothered mate etc.) and then try the comparatively advance theory (like opposition). If you are already good at endgame, then invite your opponent directly from the opening, by exchanging valuable pieces early in the game.

It's a long journey, but if you follow the aforementioned steps, you will definitely improve in a significant margin.

Best of Luck.

What do you call blitz? When I played it was zero seconds per move. Now it seems to be variable like SD1 to SD5.

At your rating it really does not matter what you do. For sure you will not ever improve if you play blitz. It is more likely to reduce your actual rating a LOT.

What you need to do is improve your skills. Then practice at playing at exactly ten seconds per move. Get in the habit of not moving instantly, but also not taking time that would run your clock out before the game ends.

You need to study tactics and learn an opening thoroughly.

15+10 is like postal chess to me.
What do you do with all that time?

Play 1+10 and force yourself to learn to move faster. You wont play much worse for a while and you will improve. But do study tactics and learn openings. Then learn end games.

Welcome to Chess StackExchange!

You say the transition from games at 15+10 to blitz sees you going from rarely losing on time to almost always losing on time.

Can I ask if the blitz is still with increment eg like 5+5 or some such?

When I switch from say bullet to 5+5 to 8+5 and such it always takes me several games to get into the rhythm of the new time control.

Do you have a sense at the shorter time controls that you are falling behind in the opening phase, middle game or end game? Are you spending too much time in the openings to get a good position but then run out of time to win?

One way to cope with the time issue is to play a small number of quieter openings. You will learn how to play their characteristic pawn structures and where to put your pieces. For example I aim to play as white the London System and as black against e4 the French Defense. Against d4 I like the NimzoIndian.

There are online games repositories with notes that can help you.

A good area to study is the endgame, an extra effort here will dramatically improve your results.

Rook and pawn endings are quite common and it's worth studying them. I find it's the area where my opponents often stumble. Always try to get your rook active even at the cost of a pawn and remember that whether in attack or defence the rook belongs behind the passed pawn!

Pay particular attention to your king safety or your opponents lack thereof. In a time scramble an exposed king position can be a nightmare.

Best of luck!