Does infinity vanish when you can go anywhere in the universe?

Everywhere isn't just the planets, there is plenty of space between them. You can't go anywhere in the universe because it is expanding. There will be places in the universe that don't exist yet.
But at some point the universe will shrink down to a black hole. Or so the theory goes! I'm not sure whether it will return to a singularity (?) but if it did you could be everywhere at the same time and infinite distance would vanish. There would at any rate be other infinities. In face some mathematicians would say there are an infinite number of infinities, a countable infinity of infinities, but there's an uncountable infinity of places.

If the universe is infinite, you never stop. If it's finite, as we currently believe, you would eventually visit every planet if you lived long enough.