I am on the edge of investing in the Sony a6000 ...

Ok, fallacy #1. :) You never invest in a camera unless you're a pro and can write it off on your taxes. Cameras depreciate. Even while new. Your "investment" will never give you any monetary returns. This is an expense, pure and simple.

(If anybody has other suggestions in that price range - I am open).

With mirrorless, I say that if the sensor is most important to you, Sony e-mount; if the system and lenses are most important to you, then micro four-thirds (Panasonic/Olympus), and if the color and UI/haptics are most important to you, then Fuji X. Don't forget that you can purchase used cameras and lenses as well.

The reason I keep straying away is the fact that the kit lens isn't very highly regarded. ...

This, my young Padawan is true in any interchangeable lens camera system. That doesn't mean the kit lens is garbage or that by its very nature it will instantly make every photo you take with it butt ugly. It is a limited lens, and there are nicer ones out there. But generally a lot of the crappy images you see taken with one are due more to beginner's mistakes with it, trying to use it for subjects outside its limits (low light), and simple bad composition from newbs.

A better camera doesn't make better pictures all by itself. It's just the tool or instrument. More expensive cameras seem to take better pictures, because the more dedicated photographer who's willing to put in more effort, 10,000 hours, and think about photography a lot more is liable to own the higher end gear, and learn the post-processing skillz, too. In other words, just because Itzhak Perlman plays a Stradivarius doesn't mean that buying a Stradivarius instantly confers on you the ability to play like Itzhak Perlman.

Chances are good you're going to be a bigger limit to image quality than the kit lens for at least a month. Maybe a year. :)

... is the kit lens good? ... the fact that it can zoom in and out is appealing.

This is your tradeoff between the kit and a prime. The kit zooms, but is slow. The prime is fast, but doesn't zoom. What do you prize more? Framing versatility? Or night time/indoors shooting? The prime can be better optically, but bad technique could easily wipe out any advantage the prime gives you. And f/8 is a huge equalizer among lenses, if you're shooting in the sunshine.

I can't really afford much more than either the kit lens, or the body and a single focal length lens.

Ok, THIS is actually what I see as your main issue. The versatility, ability, and power of a camera system is the ability to use multiple lenses. If you can't afford more than one lens, you can't take advantage of this. System cameras, by their nature, are astronomically more expensive than fixed-lens cameras. You may want to reconsider whether you actually can afford a system camera vs. getting an advanced compact instead, such as a Sony RX-100, Canon G5/7/9X, or Panasonic LX100.

If you're really jonesed about getting a fast prime you might even consider a Fuji X70 or X100.

See: What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera?

Can I take a large range of really good pictures with just a single focal length prime, and if so, what would it be? 35mm or 50mm?

You can't take a massive range of images, but you can take really good images in what is probably your most-used range with a single focal length prime. However, as a beginner, you're more liable to be frustrated with the framing limits of a prime than a more experienced user might be. In Europe, I'd probably travel with a wide-ish normal fast prime (which on APS-C, would be in the 24-35mm range), an ultrawide zoom, and a longer portrait prime. I could also take my X100T (fixed 35e prime) if I felt I had to go super light, but I'd feel more limited vs. the three lens route. OTOH, I'd also feel a lot lighter and free-er.