How long do I stay in ruku and sujud?

It depends on how fast you pray, but the minimum is a calm pause in each position. There is no fixed number of seconds. What counts is stillness, called tumaninah: you bow, stop moving, and then go on. A hurried "peck" with no pause can make the prayer invalid, so a steady, unhurried pace is best.

The minimum: be still

A man once prayed quickly in the mosque, and the Prophet ﷺ told him three times, "Go back and pray, for you have not prayed." When the man asked to be taught, the Prophet ﷺ said: bow until you are at ease, stand up straight, prostrate until you are at ease, then sit until you are at ease, and do this in all your prayers (Sahih al-Bukhari 757).

That "at ease" is the key. Many scholars describe the least you can do as a pause long enough to say "Subhan Allah" once in each position. The Hanafi school treats stillness as required too, and the Shafi'i and Hanbali schools count it as a pillar of the prayer. If you notice you've rushed a position, don't panic: do the next one calmly, and ask your imam what to do if you realise you skipped the stillness completely.

What the Prophet's ﷺ prayer looked like

Al-Bara' ibn Azib said that the Prophet's bowing, his prostrations, his standing after the bow and his sitting between the two prostrations were all about equal in length (Sahih al-Bukhari 801). In other words, nothing was rushed, and the positions were balanced. For most people, a good pace is to say the glorification three times in each position. See What do I say in ruku and sujud?

What to avoid

The Prophet ﷺ forbade "pecking like a crow", meaning dipping down and up so fast that there is no pause (Sunan Abi Dawud 862). Signs that you may be rushing:

Your forehead touches the ground and lifts straight away.
You're already moving to the next position while saying "Allahu akbar".
You can't remember the glorification you just said.

A simple fix is to say the full phrase once, silently, before you move.

When praying behind an imam

In congregation, follow the imam's pace. Don't go down before he does, and don't stay down after he rises. The Prophet ﷺ said: "The Imam is to be followed. Pray standing if he prays standing and bow when he bows; rise when he rises" (Sahih al-Bukhari 689). If the imam is fast, say the glorification at least once, calmly, and move with him. If it is a regular problem, a gentle, polite word with the imam can help. See How do I pray behind an imam?

If you're ill or elderly

If a position hurts, you don't have to stay long. Do the pause you can, and Allah accepts what is within your ability. For praying with difficulty, see How do I pray on a chair at the mosque?

Key points

There is no fixed number of seconds; the minimum is stillness in each position.
Say the glorification slowly, ideally three times.
Don't "peck like a crow" by rushing down and up.
Behind an imam, move with him and keep a minimal pause.

Tip: Count "one, two, three" in your head in sujud when you start, until the pace feels natural. Many people discover they were far faster than they thought.