To "run everything" you need a genset capable of providing 30A (3600 watts) via a single outlet. Often a genset will be rated with enough wattage, but you can only get the full watts via either a 240v outlet (which your 30A trailer cannot utilize) or by using both of a pair of 15A outlets. Make sure the genset has a 30A/120v outlet. An inverter-type generator is the best for RVers because it can throttle back to reduced RPMs when loads are light, reducing noise and fuel consumption substantially.
Gas vs propane involves trade-offs. Propane has less energy than gasoline, so an LP (propane) genset uses for fuel to produce the same watts of power, and often the LP version of a genset will be rated for lower output than the gas version. And if you want a portable genset (likely, in a small trailer), LP is not a very practical option anyway because the portable tanks are bulky and heavy. Gasoline is more widely available than LP, but you have to carry an extra container for it and carrying gas cans can be a hassle. Portable LP bottles are already designed for safe transport, but that is also why they are heavy and bulky. If you were thinking to run off the trailers own LP tanks, that is practical only for very limited use. LP gensets use too much propane to run very long from a pair of 20# or 30# bottles, and if you run out of propane you lose heat, stove and hot water as well as genset electric.