There is this driver "who would go" or "who goes"

If you want to explain a situation where, for example, you want to talk about a bus driver who goes and take a coffee sometimes on the way, which one would be more appropriate:

There is this one driver, who goes and gets a coffee on the way and you have to wait like for 10 minutes.

There is this one driver, who would go and get a coffee on the way, and you would have to wait like for 10 minutes.

To me the "would" form is appropriate either if the whole thing is hypothetical:

There might be one driver who would go and get a coffee...

or in the past:

There used to be one driver who would go and get a coffee ...

If it is non-hypothetical and present, I wouldn't use "would":

There is this one driver, who goes and gets a coffee ...

When we are telling a story, we usually use either the present tense or the past tense. (http://en.wikipedia.org/wiki/Narrative_mode) Your first sentence uses the present tense, which is fine:

There is [present existence] this one driver, who goes and gets [present habit] a coffee on the way and you have [present habit] to wait like for 10 minutes.

Your second sentence mixes the present tense and past tense, which is not fine:

There is [present existence] this one driver, who would [past habit] go and get a coffee on the way, and you would [past habit] have to wait like for 10 minutes.

So just change is to was:

There was [past existence] this one driver, who would [past habit] go and get a coffee on the way, and you would [past habit] have to wait like for 10 minutes.

(You also omitted the to from have to in your second sentence. I assume that's just a slip.)