LHR to Southampton

Flying from the US to LHR, arriving on a Thursday morning around 6am, with a cruise leaving Friday morning out of Southampton.

We have a hotel booked for the Thursday night in Southampton and I was wondering if rail or bus was the easiest way. We are probably going ti each have one carry on suitcase, one checked back, and one backpack each.

I’m trying to avoid car services and I just don’t trust them especially in a foreign country.

Thank you!