That is the large, main branch from which many smaller branches branch out.
You have a lot of questions about this sentence. Let's rearrange them so the answers build on each other.
Is there a clause or a prepositional phrase in the sentence?
There are two clauses and a preposition phrase.†
That is the large, main branch is an independent clause: it can stand by itself as a grammatical sentence.
from which many smaller branches branch out is a dependent clause: it cannot stand by itself but has to "hang" (that is what depend means) off of an independent clause. Another term for this is subordinate clause: it is "set in order beneath" the independent clause, just as the people who work under your direction are your subordinates.
is it an adjective clause ... or an adverb clause ... ?
Let's call it an adjectival ('adjective-like') clause, because, yes, syntactically it "modifies" the noun phrase the large, main branch.‡ It doesn't "modify" the verb branch out, because that verb is contained within it.
from which is a preposition phrase acting as a constituent (sub-part) of the dependent clause.
If it is a prepositional phrase, what is the object?
From is the preposition, which is its object.
The relative clause in your example is a pretty complicated construction; it may help you if I describe how it works in some detail. It is theoretically "based on" or "derived from" an ordinary independent clause, #1 below. The term we want to modify, the large, main branch, is represented in this sentence by it; to turn the independent cause into a dependent relative clause, we start by replacing it with the appropriate relativizer, which (#2) and move that to the beginning of the clause (#3). That's a grammatical relative clause; but it leaves the preposition from "stranded" at the very end, and in reading that can be difficult to parse. So to make it clearer what role which plays in this clause we "pied-pipe" the preposition along with its object, which, to the front (#4).
1. Many smaller branches branch out from it.
2. many smaller branches branch out from which
3. which many smaller branches branch out from
4. from which many smaller branches branch out
As you see, the preposition only governs which, not the rest of the relative clause; this preposition phrase is syntactically only a piece of the relative clause.
Now, the questions in your title:
Should clauses always start with a relative or subordinate? Can it also start with a preposition?
Dependent clauses are often introduced with a subordinator, which may be a relativizer, a preposition or a "subordinating conjunction".§
In many cases, however, some subordinators can be omitted, like those in parentheses in the sentences below:
That's the man (whom) I saw.
Mom says (that) I have to do my homework now.
The rules for these omissions depend on what is being omitted in what context.
Your term "start" is a little ambiguous. Independent clauses are never introduced with a subordinator—a subordinator marks the clause as dependent. But an independent clause may "start" with a constituent which is introduced with a subordinator; in this case, the subordinator only marks the following constituent, not the clause it is a part of.
[That this is complicated] is obvious!
Can there be clauses within prepositional phrases?
Sure! A subordinate clause may act as the object of a preposition,§ or a subordinate clause may "modify" the object of a preposition. In both cases the subordinate clause is inside the preposition phrase. But there is not an instance of this in your example, where the prepositional phrase is inside the subordinate clause.
† Note that I say preposition phrase, not prepositional phrase. A terminological practise I recommend is to say X phrase for a phrase whose head is an X and Xal phrase for a phrase which is X-like, which acts like an X. Note, too, that technically it is possible to describe the "particle" out as a preposition phrase, too! —but that is controversial and beyond the scope of your question.
‡ There are lots of fine technical distinctions to be drawn here, but again, those are really beyond the scope of your question.
§ Note that some modern grammarians have reclassified most of what traditional grammar calls "subordinating conjunctions" as prepositions.