I'd actually just suggest you do deadlifts and squats. Learn to do them correctly, and also work on torso strength. If it's your back that you're worried about, fix that problem because it's hindering your cycling performance as well -- your legs aren't going to put out maximal effort if your hips and back won't support it.

A few suggestions. First, I actually encourage people to focus on deadlifts first. It's a simpler lift, although it's easily done wrong. Get behind the bar and when you set up, don't try to lift from the waist. Squat down enough so you are using your entire legs to lift with. And focus on crunching your hips forward while you're bringing your knees backwards. When you work the hips, your body will respond by working the rest of the chain. It's also not damaging to your back or knees or whatever if you do it right.

Second, for squats, if you have problems with a standard front or back squat and are doing some squat variation, you are probably having a problem with your squat position. There are a lot more variables at work compared to the deadlift, but again, if you do it right, you shouldn't be hurting back or knees. That's an old wives' tale, with a few reasonable exceptions. Learn to do it right. Rippetoe, mentioned above, is a good resource to follow, and he does classes around the country. They are expensive but are worth it.

Third, as a temporary measure, you can use a trap bar (also called a hex bar, though it sometimes isn't hex shaped) to do deadlifts. It makes it easier to carry the weight differently from a bar in front of your shins, but it's still the same story -- learn to do it right with a straight bar and you'll do fine.

As for gaining strength without weights, two important concepts here. First, you have to stress your body at its current strength limit to achieve significant increased strength. Training at less than that simply doesn't bring about an adaptive response to increase strength. And second, there are a lot of muscles involved in delivering strength for a particular movement. The problem with machines (plate loaded and otherwise) is that they isolate muscles. That's their intent, but for building strength for cycling, you need an integrated result and they aren't that effective at it. An inclined leg press (sled) can be good if you have a particular problem or are recovering from an injury where you can't do free lifts, but you don't want to depend on it. Even a trap bar deducts some muscle groups from a deadlift, which means you build some strength in some muscles but not in the coordinated package you need to build cycling performance. You hear of so many cyclists who try the gym for the winter and don't come out stronger, and this is an important reason why. There have been some long threads on this forum about how to use weights effectively. I'd suggest you take a look at them so you at least can appreciate what the upside can be.