I live in bournemouth
you will need to drive a bit to go fossil hunting....kimmeridge bay or lulworth cove is great for that type of thing. less than an hour from bournemouth, heading towards weymouth way.
brounsea island you can get to from poole quay (which is a lovely place to visit), and is fab for rambling and peacocks!!
the best beach (IMHO), is alum chine as there is a fab swing park and little pool there. and the toilets are huge and, altho sandy, pretty clean (which is a massive plus).
moors valley is a great dey out.....you need to head back up the spur road towards ashley heath for that one. it is a 'free' day, you pay to park (which is pricy), but once there you have the 'play trail' thro the woods, tow huge play areas....one aimed at older children and one very sandy one for tinies (mainly under 5's). there is also a train, but that you need to pay for....altho it is not really pricy.
ooh, talking little trains.....poole park is great too. they too have a tiny 'noddy train' (as I call it), and there is a huge gus gorrila ball pit thingy, and a small ice rink which is fab to keep them amused. we took our when it first opened, just over a year ago, and they have these little penguin type things for little ones to hold onto to steady themselves. DS2 (then 4), LOVED it.
HTH