From an episode of Community:
Why don't they call it Planet Trek? [They] never go to a star. Not one episode.
-- I'm fairly sure that in at least one episode they're deliberately visiting a star before something spacey happens to it.
For my purposes, I'll define "go to a star" as ... well, "deliberately heading to a star". Hopefully there's at least one long distance trip ("There's a star exploding! Let's go watch from inside the blast range!") but the short distance trips ("I can't see anything going wrong if we fly inside a miasma of incandescent plasma.") count as well.
Going to Mars doesn't count as going to Sol, though, nor does drifting or being knocked into a star by mistake. Incomplete trips also count; if they're heading towards a star to take pictures and Q randomly puts them into a game of Minecraft, it still counts.