As for the bike... just go with something smooth, lightweight, and comfortable... possibly a hybrid since you are thinking a low-end mountain bike... but also leisure rides with the family. I say this... because I currenly have a true mountain bike (Gary Fisher) and it's great on the trails... but not so comfortable on the family rides. So... if it's just me and the kids... then I usually end up just taking my wifes bike to pull the trailer. It's more of a hybrid and much more comfy for pulling.
In my opinion, regarding the trailer... go with the Burley d'lite.
It's made for two kids, very lightweight, tons of cover for weather but plenty of ventilation for the hot days, as well as bug screens for the buggy days, folds down flat, and lasts forever. The one I bought 6 years ago still looks and performs like new. They cost around $400 new... but the resale is way higher on them than on the look-alikes.
Also... you can find a really nice one in great shape on eBay for the cost of a new "knock-off".
burley trailer, Outdoor Sports, Strollers items on eBay.com
Even though it seats two... it's actually very comfortable for one... plus... there's ton's of storage in it for food, supplys, etc. as well as some side pockets handy for the kids books, or sippy cups. When my daughters were younger... I even just attached their carseat to the inside. Worked like a charm.
Hope that helps!
Bob