I gather, but you did not say, that you plan to do all your camping in places that do not have electricity available? Many campsites have electrical power, even if only a 15A household outlet, and you don't need any battery or charger as long as that is available. Your trailer has a 12v power converter that will supply all the 12v needed and will also re-charge the battery(s) while doing so..
Assuming you will not have 120v electric (aka shore power) available, there are two questions to be answered:
(1) How many battery amps you will use each day (amp hours) and how many days you will need to go without recharging.
(2) What is the easiest and most cost effective way to recharge, assuming no external electric source is available
First the battery amp hours question:
12v light bulbs use 1-2 amps per hour, depending on size. The furnace fan uses about 5-7 amps while it is running, but usually runs only part time (depending on temperature, of course). The fridge power draw is negligible, as long as the interior light stays off. The water pump runs only for short times, so it too contributes very little over the course of a day, maybe only one amp/hour. Your stereo is something you will have to figure. Bottom line is that you can figure each light x 1.5 x hours of use and get a number of amp-hours (AH). Ditto for furnace run time and your stereo system. Throw in another 2 AH for fridge and water pump and that's the amp-hours you use every day. You need enough battery capacity to get you through 24 hours. With that modest load, you won't need much, but most people end up using more electric than they estimate, so you might want to double the estimate to have some reserve.
Batteries come in various sizes (amp capacity). A typical Group 24 size holds about 85 amp-hours (AH) when used lightly - less if a big load is applied. A Group 27 (slightly larger in physical size) holds about 105 AH. Golf cart batteries are 6V and have to be wired in pairs to make 12v, but they are high capacity and very rugged and long-lasting. A pair will store about 220 AH. In practice, you can only use about half the available AH in a battery - using more will severely stress it and greatly shorten its lifetime. So figure that your total battery capacity needs to be about 2x what you expect to use daily.
Now the charging question:
Without external power source, you can charge the battery with a generator or with solar power. You need to replace the amp-hours you used the previous 24 hours, so you need enough charging time to produce the same number of AH that you used.
With the generator, you can charge whenever you (and your neighbors) don't mind the noise. You don't actually charge with the generator in most cases - you plug the trailer into the generator and let the trailer recharge itself. This is nearly always more efficient than using the generator's 12v output, if it has one. We don't yet know the size of the charger that is built into your trailer (what year make & model trailer is it?), but it is likely in the 20-40 amp range. That means that it can put about 20 amps back into the battery in an hour (amps x hours of run time). The charge rate slows as the battery gets full, but you might reasonably expect to re-charge 40 amp-hours back into the battery over a 3 hour period. You get the idea...
Solar, of course, is available only when the sun shines and is at its peak power only when the sun is directly overhead of the panel. You may get a few hours a day of solar charging or you may get 10-12 hours in some locations. For solar, you need to guess how much direct sun you will get (don't forget tree shade) and use that to calculate the size of the panel(s) needed to re-charge the amps that were used. A 100 watt solar panel produces about 7 amps @ 13.6v (charging voltage), so you would need about 6 hours of direct sun to replace 40 amp-hours. A bigger panel (more watts) produces more amps, so needs fewer hours of sun to produce the same 40 amps.
At this point, you probably have more questions and also need to get some more data, so I'll cut the lecture short for today and see what you have to say.
In the RV Forum library in the Newcomers section there is a beginner's primer on RV Electrical Systems. You might want to read up on that.
RV Electrical System Primer