The one general thing I can say about generators is that they introduce more complexity, so there are more possible problems. These problems don’t usually happen, but there are things you can do to keep randomness down.
If at all possible, rent a generator from a lighting rental company, not a construction equipment company or Home Depot. That’s the simplest thing to do, as they will most likely only rent “inverter generators.” Inverter generators have more stable power and are better for HMIs and Kino Flos. That said, The Honda EM5000is and EU6500is are typical mid-size inverter generators (5000 and 6500 watts, respectively) and if you can find these elsewhere for cheaper, that should be okay.
If you don’t need to power HMIs or Kino Flos (or computers, really) from your generator, you might be okay with a cheaper Home Depot / construction rental type generator. With tungsten lights on this kind of generator, you may still get some noticeable flicker or inconsistent levels (Hey, weren’t we at f2.8 five minutes ago? The meter says f2 now!)
Another thing to know is that unless you get a specially-modified generator that’s not available everywhere, your power is split between two circuits, so a 2000 watt generator won’t power a 2000 watt light—only two 1000 watt lights. To get two separate 20 amp circuits, you need a 5000 watt or greater generator.
Some rental companies will supply a grounding stake with a generator that is meant to be driven into the ground and attached to the chassis of the generator. This generally shouldn’t be used if the generator is your sole source of power and is insulated from the ground (like with rubber or plastic wheels). If you’re running power from a generator inside a building with its own electrical system, things get a bit more complicated in the grounding arena. In that case, the generator has to be grounded to the building’s ground (Harry Box, Set Lighting Technician’s Handbook, 3rd Ed., 417-419).
And then there’s noise. Generators make a lot of noise. The smaller the generator (unless it’s a full-size film production-specific tow-behind or truck-mounted generator) the quieter it is. The farther from away, the quieter it is. Ideally, you place the generator far away behind a building far away, but that means lots of extension cords and some amount of voltage drop. If the generator has to be closer to the set, you can minimize the sound by placing the generator on soft ground (grass, not concrete) and by hanging sound blankets on C-stands or other things between the generator and the set. The more the better. At minimum, one right next to the generator (not touching it). Other places to hang sound blankets are between the generator and anything sound might bounce off of, like a building, and as close to the microphone location as possible without getting in the way of the frame, lighting, or action. Never put a sound blanket on top of the generator.
There’s a really good discussion of most aspects of generator use in Set Lighting Technician’s Handbook, by Harry Box that is worth reading if you’re doing this regularly. There’s also an excellent article online called Portable Generators in Motion Picture Production from Screen Light and Grip in Massachusetts that goes into a crazy amount of detail and also describes the above-mentioned specially-modified generators.