We finally get a moonbase up, and with that pesky "most of the weight of our spacecraft is in fuel" gone, space tech begins to advance at amazing rates. In just a few years, there's hundreds of companies up there building dozens of spaceships per day. They're even reasonably cheap... the sort of thing a group of a friends could afford if they were interested.
These ships are super easy to pilot - if you can handle using a computer and you know how to drive a car, you're good to go. The ships come with advanced AI including great mapping software, all the math is done for you.
Best of all, they're super safe! They're radiation shielded, impact shielded, and even lightly armed with advanced targetting to blow up any minor incoming space junk automatically. All systems are redundant, tons of spare parts are stored on board, and everything is color-coded and well-documented, with automatic damage sensors everywhere.
Unfortunately, no one's solved that whole "can't go much faster than .1 C" thing yet, and no one's found a cure for aging yet either. There's some great worlds out there, but it's your great grandkids that will be exploring them.
I wish we DID have a way to go over light speed. (Yes, I know this is utterly impossible by any conventional means. Let's assume scientists found a way around the limit.)