Moreover:

Пойду купаться в озере, идешь со мной?

is okay if you're speaking to a single person as opposed to a group of them or if it's obvious from the context who is the ты

Also:

Завтра в Москву уезжаю я and завтра в Москву уезжаю

Both are okay and you definitely can omit я in there. The only difference between the sentences is that the Я variant is a typical example of the speaker wanting to explicitly specify something. In that case it's я. Well, of course a proper intonation stress on я is required

An English equavalent to this is a sentence like this:

It is I who goes to Moscow tomorrow