Should I wait before I start speaking Spanish?

Wait a little, but not long. A short period of mostly listening helps you build a feel for Spanish, but you don't need months of silence before you open your mouth. My take is to start speaking once you can follow simple audio, even if that means talking to yourself.

Where the "silent period" idea comes from

Linguist Stephen Krashen, who also came up with comprehensible input, noted that many people go through a quiet stage when learning a language. They take in lots of language before they begin to speak, much like children do.

Some learners and methods take this further and say you should avoid speaking until you can understand a lot. Others say speak from day one. Both sides have a point.

The case for waiting

You hear correct models first. If you start speaking before you've heard much Spanish, you may build bad habits and fix them later.
Less pressure. Not forcing speech can reduce stress and let you enjoy learning.
Better understanding. Speaking depends on what you've absorbed, so more listening gives you more to say.

The case for starting early

Speaking is its own skill. Understanding Spanish doesn't automatically let you produce it. Plenty of people understand a lot and freeze when it's their turn. See understanding Spanish but not speaking it.
It reveals gaps. You find out what you can't say, which tells you what to learn next.
Fear grows with delay. The longer you wait, the scarier it gets.

My middle path

Here's what I'd actually do:

First few weeks: mostly listening and reading. Repeat phrases quietly or out loud to practice sounds.
Once you can follow simple Spanish: start talking to yourself. Describe what you're doing, answer questions from the audio, say your day.
A bit later: try short conversations with a tutor or exchange partner. Keep them simple and don't aim to be perfect.

During the listening-first weeks, you want input you can actually follow. Full disclosure, I built Langtok for that: short Spanish stories where each word highlights as it's spoken, and you can slow them down to 0.75x if the speech is too quick.

You don't need to be ready. You need to be able to say a few things and be willing to say them badly. A sentence like Hoy hice ejercicio y luego cociné (today I exercised and then I cooked) is a fine thing to practice.

What to do about fear

If speaking makes you nervous, start in the safest setting. Alone, then with a patient tutor, then with a stranger. Fear of speaking Spanish covers this in more detail.

Bottom line

A short listening-first phase is smart. A long silent phase usually isn't. The goal is to be listening a lot and speaking a little early, and then slowly speaking more as your Spanish grows. Don't wait until you feel ready, because you almost never will.