How much grammar do you need to speak Spanish?

You need a lot less grammar than most courses teach to start speaking Spanish. The present tense, ir a + infinitive, a rough past tense, ser and estar, and a small set of pronouns will carry you through many everyday conversations. The rest builds up slowly, mostly by hearing it used, not by studying it.

The core you actually need

If I had to pick the grammar for your first few months, it would be this:

Present tense of regular verbs and the common irregular ones (ser, estar, tener, ir, hacer, querer, poder)
Ir a + infinitive for the future: Voy a trabajar (I'm going to work)
A basic past: the preterite for things that happened, even if you make mistakes. Ayer comí con mi familia (Yesterday I ate with my family).
Ser* vs *estar at a basic level (see ser vs estar)
Gender and agreement: la casa blanca, el libro blanco
Common patterns: me gusta, tengo que, quiero, necesito, hay
Question words and negation: ¿Dónde está?, No entiendo, No tengo

That's not a complete list of Spanish grammar. It's the part that gets you talking.

What can wait

Plenty of topics sound scary and don't need to be perfect early on:

The subjunctive (you can use a few phrases like quiero que and ojalá before you understand the whole system)
Perfect tenses (he comido, había comido)
Conditional and past subjunctive (si tuviera, haría)
Perfect command forms, passive voice, and obscure connectors
Vosotros, unless you're heading to Spain

If you're curious, when to use the subjunctive is a good reference when you're ready.

Does grammar even help?

Yes, but it has a smaller role than people think. Grammar knowledge helps you notice patterns and fix errors, but it doesn't make you fluent on its own. You get there by hearing and reading lots of Spanish and by speaking. This is the idea behind comprehensible input: your brain picks up patterns from examples you can follow.

I think of grammar as a map, not the trip itself. A quick look at a map is useful. Staring at it for hours isn't.

Short stories are a decent place to see those patterns in use. I'm biased since I built Langtok, but the idea is that you swipe through Spanish stories, tap any word to translate it, and meet grammar in sentences you can follow.

Common ways to overdo it

Studying one tense for weeks before moving on
Doing hundreds of fill-in-the-blank drills without ever listening to a real person
Waiting to speak until you're correct. You will be wrong a lot. That's fine, and it's how everybody improves.
Memorizing exceptions that you can look up when you need them

A simple approach

Learn the core set above, one piece at a time, with examples.
Spend most of your time on listening and reading at a level you can mostly follow.
When you notice something you keep getting wrong, look up that specific rule.
Keep a short list of "grammar I'm fuzzy on" and revisit it every few weeks.

That's it. Honestly, I think most learners would do better by spending half their grammar time on listening instead. If you want to know what comes first, grammar or vocabulary has a good answer.