You should start by estimating the power draw of all your appliances, and then based on how long you want to be able to camp between charging your batteries you can figure your battery capacity requirements. I think Adventure Trailer's web site has some good info on this subject. Also, check out the latest issue of Overland Journal for some good data on power consumption for popular fridge models. IIRC, you are looking at about 1.7 AH average power draw for a fridge. The power draw for your fridge may vary from that depending on the model you have.
A good quality group 31 AGM battery will typically be rated at about 105 AH, so that should give you about 60 hours of use with just the fridge, so about 2-1/2 days without recharging the battery.
I use stick up LED lights in my trailer but they each have batteries in them rather then using the trailer battery. It's a little more maintenance to keep fresh batteries in them, but I usually travel with spare batteries for flashlights etc., and it saves a little draw off the main trailer battery.
A fairly common trailer set up is two 12v Group 31 batteries in parallel for more capacity. You can get some capacity benefit here, but often not the doubling you'd expect. It can also cause lots of issues if the internal resistance of the two batteries is different. The common recommendation when running two batteries in parallel is to always use two identical brand/model batteries with the same date code to try to minimize any difference in the electrical characteristics between them.
I used to have the two 12v battery in parallel setup in my camper and had no end of problems with it, so I converted it to a single 8D size battery that has about 200 AH of capacity. I've had better luck with that setup, and I'm currently thinking of converting my trailer over from dual Group 31 12v batteries to a single 8D 12v battery. Same capacity, same weight, and gets away from the parasitic discharge issues of two batteries in parallel. The challenge is you need room for one large battery (Group 8D is about 21" x 9" x 9").
Another alternative to consider that may give you more capacity while avoiding the issues with two 12V batteries in parallel is to use two 6V batteries in series. It's a good option if you have room - 6v batteries tend to be tall. Be sure to check on the charging voltage requirements though. You may find you need over 14v to fully charge a bank of two 6v batteries in series, and typically your voltage regulator in your vehicle won't go that high. When thinking about charging voltage you also need to consider the voltage drop in the wiring back to the trailer.
