You can use many conjunctions in a sentence; but you have to use them the right way.
It's complicated, but here's a simplified explanation which handles the example you offer.
At clause level there are two sorts of conjunctions, which establish different relationships between the clauses they join.
a coordinating conjunction (Collins, so, 18-21 calls this a sentence connector) puts the two clauses it joins on the same ‘level’: each clause is syntactically independent.
Just for example, let’s suppose that there’s a network in which the members are obliged to pass on interesting news.
A so B
Jim told Agnes, so of course she told me.
In theory you can chain as many coordinating conjunctions as you want; the only limit to the number of clauses you can join this way is your hearer’s patience.
A so B so C so D so E ...
Harry told Louise, so she told Jim, so he told Agnes, so she told me, so I will tell Phyllis, so she will tell ...
This is not good written style, but it is very common in oral storytelling.
a subordinating conjunction puts the clause it introduces at a ‘lower level’ than the clause it is joined to. That is, the clause introduced by a subordinating conjunction is not independent: it plays a distinct role within the other clause.
Agnes told me because Jim told her.
because Jim told her explains why Agnes told me.
Again, there is no theoretical limit on how many subordinations you may chain
because D ...
Agnes told me because Jim told her because Louise told him because Harry told her, because ...
Each clause explains the clause ‘above’ it.
Now: you are free to combine coordinating and subordinating clauses to express complex relationships, as long as you keep the relationships straight.
A so C
Jim told Agnes because Louise told him, so Agnes told me.
But: what you cannot do is put two clauses in two different relationships to each other; and that is what your example does:
∗ Because it is raining today, so I go out with my umbrella.
A so B
When you use both a subordinating and a coordinating conjunction, A shows up twice in the diagram. The meaning is clear, but syntactically A is assigned two conflicting roles, which is a no-no.
Just by the way: your construction was acceptable in older English, down to about the middle of the 17th century; but it disappeared in the ‘rationalization’ of formal discourse which occurred after the Restoration.
Note, by the way, that although because is always a subordinating conjunction, so plays different roles in different contexts. In the examples above, it is a coordinating conjunction; in other cases it may act as a subordinating conjunction:
Because it is raining today I’ll take my umbrella so I don’t get wet.
English, as I’m sure you’re aware, is full of traps like this.