Okay, I'm sure it's obvious, but my camera knowledge is absolutely *zilch*.

I didn't realize I was asking for 3 different lenses! So, will any of these lenses "come with" the camera?
First pointer - no matter what camera you are using, if you really seriously want to get better photos, nothing at all can replace learning a bit about the basics of photography. That's not to say there aren't better cameras or worse cameras. But when it comes down to it, a good photographer can still take good photos even with a bad camera, and a bad photographer can still take bad photos even with a great camera! You don't need to become a great photographer - but like anything else (computers, cars, cooking, you name it), you can get better at it with practice and a little learning.
I only say this because I see a lot of people who want to upgrade cameras so they can start taking marvelous photos in all these different situations, but without realizing that the camera isn't the only reason they don't always get the results they want. Knowledge and skill will help improve your photos much much more than a new camera will. Depending on what 'Cybershot' model you have now, you may actually have a camera with much more ability than you realize (Cybershots are a brand name...there are hundreds of actual models of Cybershots, ranging from basic pocket cams to serious prosumer and ultrazooms).
I'm most interested in the night shots. Of all our trips to Disney, we've NEVER been able to take pics at night
(or of any indoor shows.)
Our Sony is decent if the lighting is just perfect, but other than that it's useless to us!

We could keep it for the basic outdoor/daytime pics. I guess I don't have to have the action shots right now.
Actually, ALL digital cameras, even the cheapest ones, can pull of decent night shots. Some are much better than others, but when it comes to those lovely lights and smooth waters and crisp details and colors that most people like in night photography, a mid-grade or higher P&S model will take night shots almost as well as a high-end DSLR. Beautiful night shots only require a long exposure...so your camera's longest shutter speed is the only factor. Cheap pocket P&S models are sometimes limited to only 1 second shutter speeds, so those will give you passable but not great results. Mid-line pocket P&Ss, and any high-end P&S, will usually give you up to 30 seconds...and with those, you'd be hard-pressed to tell the difference between those and a multi-thousand dollar DSLR shot. Depending on which Cybershot model you have, you may actually be capable of some pretty stunning night shots with your current camera! It all comes down to the technique.
Check out this gallery of my night shots...the first 92 photos in the gallery were all taken with 3 different Sony Cybershot models...a 2003 Sony F717, a 2005 Sony H5, and a 2007 Sony T100 ultra-compact. The rest of the photos were taken with a DSLR. Personally, I think the best from the Cybershots compare quite well with the best from the DSLRs:
http://www.pbase.com/zackiedawg/night_photography&page=all
Now, of the shots you mentioned, your current camera may be just fine for night shots (depending on the model you have). Even if it is a basic model with only 1 second shutters, you can still get much better night shots than you are currently getting just by following some basic rules, like keeping ISO to 100 or 200, slow shutter speed, open aperture, and most importantly, put the camera on a level surface and don't touch it - use the self-timer and let it take the photo with your hands off.
Where your current camera will not be able to keep up with a DSLR is with the action shots and ride shots. But remember that a DSLR won't cure your problem immediately - much of the ability of a DSLR to shoot action and indoor scenes comes from the lens you are using. Most cheap DSLRs will come with a cheap, basic lens...it will be plenty usable - and at least as good as your P&S in most situations...but it won't be as light sensitive, and therefore will force you to use very high ISO to compensate. And though a DSLR can shoot at much higher ISO than a P&S and still be good quality, you will suffer from that 'grainy' or 'noisy' look as you push higher. Most of us with DSLRs that want to shoot action or low light end up spending money on low light lenses or fast action lenses...and they cost much more money!
A DSLR is ultimately the most flexible and best type of camera to get. But like any other top of the line tool, it has its compromises that users accept to get that felxibility and quality. They're much heavier, bigger, more expensive, and generally require more maintenance. And that maximum flexibility comes at a price...namely the cost of adding the lenses you need for each scenario. I've always felt it was probably best to stick with a good P&S, or a basic DSLR and kit lens, while you learn more about photography and camera control, and only when you've really begun to grasp camera settings and understand how to set up for different kinds of shots, should you then go out and spend bigger money on the serious lenses.
Hope that helps a bit!