When proving a biconditional, how to say the "I'll prove left-to-right first" in words?

When proving a biconditional, sometimes the writer says "I'll prove the converse first". I want to say I'll prove left-to-right first. How to say that? I don't want to put a left to right arrow. I prefer words.

Usually I just see "forward implication" ($\implies$ in symbols) for the "left to right" implication, and "the converse" ($\impliedby$ in symbols) for the "right to left" implication. These seem clear enough, so I doubt anyone would have issue with you using those phrasings.

Also, how can I say I'll prove the converse of the statement but my strategy is to prove the contrapositive of the converse. Should I just say "I'll prove the contrapositive of the converse"?

I don't really have a proper answer for this one, though, but I feel that your phrasing works just as well.

When proving a biconditional, sometimes the writer says "I'll prove the converse first". I want to say I'll prove left-to-right first. How to say that? I don't want to put a left to right arrow. I prefer words.

Eevee Trainer suggests "forward implication" in their answer, and I think that sounds good: "I'll prove the forward implication first." The opposite, of course, would be "I'll prove the reverse implication first."

An alternative which I think also sounds good is to say "forward direction" instead of "forward implication."

Also, how can I say I'll prove the converse of the statement but my strategy is to prove the contrapositive of the converse. Should I just say "I'll prove the contrapositive of the converse"?

The contrapositive of the converse is called the inverse. Don't call it the contrapositive of the converse, since that sounds redundant (just like if you called it "the inverse of the converse of the converse").

Putting all this together, you might write something like: "First, I'll prove the forward implication. Next, I'll prove the inverse of the forward implication."