Congrats on the new trailer and welcome to the RV world. As for storage times take your propane tank and battery with you. Not complicated to do. Traveling with a full tank should not be a problem but as mentioned it takes away from your cargo carrying capacity so you have to be mindful of how much 'stuff' you take with you. If you plan on staying at campgrounds that have full hookup then only fill it with about 5 gallons, enough to flush the toilet a few times if you need to while traveling. It beats using the gas station bathroom.
Being new you have a few things to learn. Tank management especially when not hooked up to sewer can be critical. It can fill up fast and maybe force you to pull over to the dump station sooner than planned, a little inconvenient but not overwhelming. Dump only when the tank is 3/4 or more full. Obviously if its not full when you are ready to leave dump anyway using LOTS of water. Use a septic compatible tissue too. It desolves easier. Get a clear elbow for the dump takeoff so you can see when the solids are out of the tank, kinda gross but necessary.
Have fun learning and enjoy the process. This forum is here to help so don't hesitate to ask questions.