When to use preposizioni "A" vs "In", and also when do you conjugate? Such as "Stasera andate al cinema, vero?" I was under the impression "in" was used when talking about places. Another question pertaining to the above sentence is, why is it "al" and not "a", what signal is given in a sentence that says you need to conjugate?
I'll base my answer on what is explained by Federica Colombo in the book Grammatica e pratica della lingua italiana per studenti stranieri (edizioni ELI, 2006).
I'll begin with your last question. As explained by @DaG, this is not called "to conjugate", but "preposizioni articolate". A "preposizione articolata" consists in a "preposizione semplice" (such as "a" or "in") put together with a determinative article so as to form a single word. For prepositions "a" or "in" (but also for "di", "da" and "su"), one uses "preposizioni articolate" whenever the noun that follows the preposition is preceeded by a determinative article, following these rules:
a + il ---> al
a + lo ---> allo
a + l' ---> all'
a + la ---> alla
a + i ---> ai
a + gli ---> agli
a + le ---> alle
in + il ---> nel
in + lo ---> nello
in + l' ---> nell'
in + la ---> nella
in + i ---> nei
in + gli ---> negli
in + le ---> nelle
Note that "in" is transformed into "ne" whenever it forms a "preposizione articolata".
To give some examples, we don't say "vado *a il mare", but "vado al mare"; we don't say "vado *a lo zoo", but "vado allo zoo"; we don't say "vado *a l'università", but "vado all'università".
Let's consider now the question about the uses of "a" versus "in". It's impossible to list all possible uses of these prepositions, but one can explain the main ones according to Colombo's book. As you mention in your question, preposition "in", but also preposition "a", are used when talking about places. The use of "in" versus "a" in complements of place is largely addressed in the answers to this question and this other question, so let's explain other important uses of these prepositions.
Preposition "a" is used to express an indirect object. For instance,
Restituisci la macchinina a Stefano!
Preposition "in" is used to express the means of transportation:
Vado a Roma in treno.
Both these prepositions can also be used to express time complements:
- Preposition "a" is used in espressions such as "a domani", "a presto" and before the names of festivities and times of the day, as in these examples:
A Ferragosto le città italiane sono deserte.
Ci vediamo a mezzogiorno.
L'autobus parte alle tre.
- Preposition "in" is used before the names of months (however "a" is also used) and seasons, and also to express the amount of time needed to do something (some examples are taken from this book):
Quest'anno andiamo in vacanza in settembre.
In inverno vado spesso a sciare.
In autunno piove spesso.
Da Bologna a Milano si va in due ore.
Note that "in" followed by the name of a season can also be expressed as "di" followed by the name of the season; for instance, "in inverno" = "d'inverno".
Both these prepositions are frequently used in lots of fixed idiomatic expressions, as in the following examples.
Preposition "a" is used in some fixed espressions which express the way something is done or the characteristics of something, such as "a piedi", "a cavallo", "a righe", "a quadretti", etc.
Preposition "in" is used in fixed expressions such as "in tempo", "in ritardo", "in anticipo", etc.