The passato prossimo is found all over the place and is easy to use, although of course there are some irregular forms and you have to be careful about the minority of verbs that take essere instead of avere as an auxiliary. The passato remoto presents more difficulties.
First, let's look at the conjugation of the most important verbs, to be and to have. They are irregular. You should memorize these.
Essere:
io fui
tu fosti
lui/lei fu
noi fummo
voi foste
essi furono.
Avere:
io ebbi
tu avesti
lui/lei ebbe
noi avemmo
voi aveste
essi ebbero
Notice something? The forms fosti, foste and avesti, aveste are similar and easy to mix up. They are also similar to fossi and avessi, which are not even forms of the passato remoto, but imperfect subjunctives. This is a good time to remind you of the highly useful presence of online verb conjugators, where you can plug in the form and see what it is. Or plug in the infinitive and get all the forms. Here is Coniuga.
Here are the three regular conjugations of our old pals -are -ere -ire:
- io parlai
- tu parlasti
- lei/lui parlò
- noi parlammo
- voi parlaste
- essi parlarono
- io ricevetti (or: ricevei)
- tu ricevesti
- lei/lui ricevette (or: ricevé)
- noi ricevemmo
- voi riceveste
- essi ricevettero (or: riceverono)
- io dormii
- tu dormisti
- lei/lui dormí
- noi dormimmo
- voi dormiste
- essi dormirono
Isn't this BORing?
Now, I'm going to tell you something you'll like. There is an active knowledge of language (speaking and writing) and a passive knowledge (listening and reading). Generally, the latter is easier. Now, the passato remoto is squarely in the passive area for foreigners, and even for most Italians. It is rarely used in speech (sometimes by Southerners, which I always find cute) and mostly used in written Italian of a more formal nature (published works). So you will see it when you start reading a lot of "serious" stuff in Italian. If you do, you will have to recognize the form.
Usage: it is an oversimplification to say that the passato prossimo is simply more distant in time than the passato prossimo. The perception of time is subjective and even cultural. But for all practical purposes, you, learner of Italian, will not have to use it in an active way.
Interesting bit of trivia: the form fu can also be used as "late" i.e. dead, as in Pirandello's novel Il fu Mattia Pascal.
