I kind of agree that, if we only care about effective communication using valid Spanish (and not about actually speaking good Spanish), the number of tenses can be greatly reduced.
Mind you: doing so will obviously slightly change the tone and nuances and other small details of what you are saying, but you will be understood nonetheless.
If I had to say which tenses... No wait, in fact I have to, right? That's the point of the question.
Non-finite forms
Impersonal or non-finite forms are like basic building blocks. You need them to form compound tenses and phrasal verbs. You can't do much without them.
- Infinitive, gerund, (past) participle → must learn
Technically, if you only care about effective communication, you could stop here and just speak like a Hollywood Indian:
Peter Pan es un valiente guerrero. Ayer salvó a Tigrilla e hizo al Gran Jefe feliz. Mañana se casará con Tigrilla. → Peter Pan. Valiente guerrero. Ayer salvar Tigrilla. Hacer Gran Jefe feliz. Mañana casar con Tigrilla.
But that'd be going too far in our quest for simplification, IMHO (and probably embarrassing to you and to anybody listening).
Indicative
Indicative is the mood used to talk about the real world. Things you do or did or will do; events in the past, present or future. You need to learn at least the simplest tenses: it will make it easier for people to understand you, it will make your speech sound more natural, and they're needed to replace more complex tenses and moods.
Simple tenses
- Present → must learn.
Imperfect → must learn. When using action verbs (correr, saltar, subir...), imperfect of those verbs can be replaced with imperfect of estar + gerund; still need to learn it for situations when action verbs are used for facts rather than actions, for any other verbs, and to replace the conditional, too.
La niña corría hacia la tienda. → La niña estaba corriendo hacia la tienda.
Me caí cuando subía por las escaleras. → Me caí cuando estaba subiendo por las escaleras.
El autobús no pasaba por mi barrio. → ❌El autobús no estaba pasando por mi barrio.
Juan tenía 200 pesos en su bolsillo. → ❌Juan estaba teniendo 200 pesos en su bolsillo.
Lucía sabía la respuesta. → ❌Lucía estaba sabiendo la respuesta.
Preterite → must learn.
- Future → can be replaced with present + infinitive, or just present (adverbs help here). Actually there's some debate on whether the future tense is bound to disappear.
Mañana cenaré con Noa. → Mañana voy a cenar con Noa.
Mañana cenaré con Noa. → Mañana ceno con Noa.
Compound tenses
Present perfect → can be replaced with preterite.
He ido a comprar. → Fui a comprar.
Este mes ha llovido mucho. → Este mes llovió mucho.
Past perfect → good to learn. Often times it can be replaced with preterite; other times with estar/llevar + gerund/infinitive. Some other times you might need to discard the main verb entirely and use some alternative form; it may be easier to just learn the tense.
¿El libro no lo habías cogido tú? → ¿El libro no lo cogiste tú?
A las 10, la obra había terminado. → A las 10, la obra estaba terminada.
Cuando llegué, habían hecho una gran parte del trabajo. → Cuando llegué, llevaban hecha una gran parte del trabajo.
Había esperado varias horas antes de entrar. → Estuve esperando varias horas antes de entrar.
Todavía no había empezado a comer. → Todavía no estaba comiendo.
Ya había conseguido el pase. → Ya tenía el pase.
Past anterior → can be replaced with preterite.
Cuando hubieron llegado todos, empezó la ceremonia. → Cuando llegaron todos, empezó la ceremonia.
Apenas María hubo terminado la canción, su padre entró. → Apenas María terminó la canción, su padre entró.
Future perfect → the auxiliar verb can be replaced with present ir a + haber.
Cuando venga, ya habrá terminado. → Cuando venga, ya va a haber terminado.
Imperative
The imperative mood is used to give orders and commands. However, given that current social norms favor politeness, this mood is seldom needed, as there are "nicer" ways of telling other people to do things. It has only one tense and it can be easily replaced.
- Present → can be replaced with present ind. (by turning it into a request or by throwing some adverbs into the mix), or even with just an adverb, in some cases.
Recoge eso. → ¿Puedes recoger eso?
Vete a tu habitación. → Ahora te vas a tu habitación.
Salid de ahí. → Fuera de ahí.
¡Cállense! → ¡Silencio!
Subjunctive
Used for wishes, emotions, alternative possibilities... When talking about hypothetical "if ..., then ..." scenarios, subjunctive is used to talk about the "if" part. Also used in subordinate clauses. Spanish uses it a lot more than English does, so you will need to learn at least the simple tenses.
Conditional
Conditional is not a mood, but a tense. When I was a kid, it was taught as one of the indicative tenses; nowadays there seems to be some debate about that classification, so I just put it here.
When talking about hypothetical "if ..., then ..." scenarios, conditional is used to talk about the "then" part. It is also used in courtesy formulas.
Simple conditional → can be replaced with present for requests/courtesy (adding a courtesy formula like por favor to compensate for the rudeness of using present tense), and with imperfect (or imperfect + infinitive) for most anything else.
Querría un vaso de agua. → Quiero un vaso de agua, por favor.
¿Podría darme eso? → ¿Puede darme eso, por favor?
Si tuviera dinero, viajaría a Sudamérica. → Si tuviera dinero, viajaba a Sudamérica.
Yo que tú, lo olvidaría. → Yo que tú, lo olvidaba.
De pequeño, pensaba que sería médico → De pequeño, pensaba que iba a ser médico.
Compound conditional → can be replaced with present, as long as you also replace the bounded subjunctive as explained above. Bonus: doing so clears the confusion between 1st and 3rd person habría.
(1st) Si hubiera tenido ocasión, se lo habría dicho. → Si llego a tener ocasión, se lo digo.
(3rd) Si hubiera tenido ocasión, se lo habría dicho. → Si llega a tener ocasión, se lo dice.
Él no habría respondido así si tú no le hubieras insultado. → Él no responde así si tú no le insultas.
There may be some cases where the above replacements just won't work and you will have to get creative (like e.g. "No me hubiera gustado estar en su lugar" → "No me habría gustado estar en su lugar" → ...? → "Mejor no haber estado en su lugar"), but for the rest 95% they are enough.
TL;DR
- Tenses you cannot avoid: present, imperfect and preterite of the indicative mood; and present of the subjunctive mood.
- Tenses that will make your life easier: past perfect of the indicative mood, and imperfect of the subjunctive mood.
- Tenses you can survive without: future tenses, conditional tenses, any other compound tenses, and the whole imperative mood.