Draining black/gray tanks on trialer
Your dump port will likely be 3", bayonet mount. It is a good idea to have a small clear section between the trailer and the hose so you can monitor what is happening throughout the dump cycle. A 45 degree fitting worked for me on my trailer; a short straight fitting seems more appropriate for the motor home.
Then you need hose. 10' will work in most cases, but you may want another 10' section just in case (with whatever is needed to connect the 2 hoses together). For the far end, a 'universal fitting' is the best. This is an angle, which has several threaded sections of different sizes. I like the 'blue' one (Prest-O-Fit?). Disposable gloves and/or disinfectant are handy to keep yourself safe.
There are 3 ways to 'flush' the tank.
1) Using a wand down the toilet. This is cheap and effective, but you have to run a hose from somewhere to the bathroom to power the wand. I've seen kits which mount a water source in the bathroom, and I would probably consider this if I used this method. The time I tried it, I used one of those 'chocks' you can get at the same place as the wand (Camping World for my setup) to hold the toilet open while I was wanding it.
2) Spray water back up the output. The way to do this is to have a fitting between the trailer and the hose which has a hose fitting to spray water into the system. This fitting would be in place of the 45 degree fitting mentioned above. There are 2 that I've seen. One sprays water back up the tube; this cleans the best (but only if you have a straight run from the port to the tank) and the other has another valve outside the water input so you can backfill the tank. Both these are useful, so I built a combo unit which does both
3) Probably the best and easiest to use is an internal sprayer. The down side of this is you have to drill a hole into your black tank. There is a new one out which rotates inside the tank; if I was in the market for one of these, I'd look into that one first.
Gray tanks don't have the problems with solids that black tanks do. They can have grease build ups though. If you can backfill it, do so occasionally; otherwise you can deal with most problems with appropriate chemicals.
One way to dump is to first make sure both black and gray tanks are at least 1/2 full. If not, add water if practical to get to that point. Hook up the hose, double check that everything is connected and held in place at the ends, then pull the black tank valve. Spray and/or backfill as you can, then dump again. Repeat until the dump is clear. Then close the black tank valve a final time and pull the gray tank. Backfill if you can, then dump again. Disconnect the fitting from the trailer and lift it so anything left in the hose is forced into the sewer. Disconnect and pack everything away for next time.
You don't need to spray/backfill every time, but the more often you do it, the less the chance of having tank problems.
By the way, the hose the dealer gave you is most likely the cheapest, flimsiest hose possible. You will want to replace it with the best hose you can get, as soon as possible. The 'red' system seems pretty good, but you can pretty much only use their parts and accessories to attach to the hose (the hose is 'reverse threaded'). The 'blue' system is more versatile, but I'm not thrilled with the way it attaches to the trailer (everything else about the system is great). The best hose out there is the 'Rhino' hose, but you will need to find and attach your own fittings, since it comes as raw hose. I'm partial to the green 'Level-upp' flat hose system, but good luck finding it (it was a challange to get mine 4 years ago).