I would say that the Note would be too small for 2 adults, 3 kids plus bikes in the boot; probably the next size up (C-Max size) - if you're looking for petrol versions for mostly short trips, then I would suggest (given the price range) in looking for an older Japanese estate or MPV (mainly Honda, Toyota and Mazdas, in that order) or a slightly newer Ford, which may be less reliable, aren't that bad really on that front (stay away from models with all the gizmos and tatty ones [which probably haven't been looked after]) but can be fixed relatively cheaply everywhere.

For the price, condition will be key. Give whatever you like the look of from a visual inspection a good test-drive, including giving it a bit of welly to see if the engine is ok, and driving on poorly surfaced roads/over road humps in case there are suspension problems (you'll hear odd noises if you do). A FSH, even if its not always at a main dealer (though better) is always a good indicator of a well-cared-for car.

Best of luck.

Ford focus would once again fit the bill but again the powershift box is like the DSG and less the smooth at low speeds.

One off the wall is the BMW 5 series which with a few miles on the clock be had a very reasonable prices. Loads of room, great Auto gearbox.Maybe hard to find a petrol but with deisel less than petrol at the pumps now (have a look!) may be a good option. Reliable and cheep to run