Which Spanish words should you learn first?

Learn the most frequent words first: the small verbs (ser, estar, tener, ir, hacer, querer, poder), pronouns, question words and little connectors like y, pero and porque. Those few hundred words show up in almost every sentence, so they pay off faster than any themed list of animals or colors.

Why frequency beats topics

Spanish, like every language, has a small core that does most of the work. A handful of words appear in nearly every conversation, and then there's a very long tail of words you'll see once a month.

Textbooks often organize vocabulary by topic: family, food, the house. That's tidy, but it means you learn la cuchara (the spoon) before you learn aunque (although). I think that's backwards. You can describe a spoon by pointing at it. You can't build sentences without the glue.

The core to start with

Here's what I'd put first, roughly in order of usefulness:

Verbs: ser (to be, permanent things), estar (to be, states and places), tener (to have), ir (to go), hacer (to do/make), querer (to want), poder (to be able to), saber (to know), decir (to say), ver (to see).
Pronouns: yo, tú, él, ella, nosotros, me, te, lo, la, se.
Question words: qué, quién, dónde, cuándo, cómo, por qué, cuánto.
Connectors: y (and), o (or), pero (but), porque (because), si (if), cuando (when), también (also).
Little words: no, sí, muy (very), mucho, poco, aquí, allí, ahora, hoy, siempre, nunca.

If you're not sure when to use the two "to be" verbs, ser vs estar is worth a read early on.

Learn them in phrases, not alone

A word like tengo is much easier to keep when you learn it as tengo hambre (I'm hungry) or tengo una pregunta (I have a question). Chunks also teach you grammar without you noticing. Useful starter chunks:

No entiendo (I don't understand)
¿Puedes repetir eso? (Can you repeat that?)
Quiero un café (I want a coffee)
¿Dónde está el baño? (Where's the bathroom?)
Me llamo... (My name is...)

Don't forget the free ones

If you speak English, you already know hundreds of Spanish words without trying. Importante, hospital, animal and restaurante are nearly the same in both languages. There's a whole page on cognates, and they let you spend your study time on the words that don't transfer.

What to do with this

Don't try to memorize a 1,000-word list in week one. A better routine:

Learn the verbs and connectors above in short sentences.
Add 5 to 10 new words a day from things you actually read or hear. In Langtok, the app I built, you can tap a word in a short story, save it and quiz yourself on it later.
Review old words on a schedule so they don't vanish.

After a few hundred words you'll notice something nice: simple Spanish starts making sense, and every new thing you read teaches you a few more without effort. At that point, content you enjoy matters more than any word list.