*Check campsite facilities, as some don't have showers/toilets.
Anyway, find Brockenhurst, New Milton and Lyndhurst – there's an ex railway between with various paths leading off, I've done BH to RW. *Note that this no longer legal and frankly speaking the forest for qwits have gone off one one with restrictive access. (Cycle shops will have current maps).
For family cycle Brockenhurst is a good strat – you can nearly get to Beauliue off road. They have cunningly created several unjoined sets of cycle trails Linwood, Burley Lodge, Brockenhusrt LHS A337, Brockenhurst RHS A337 – biggest concentration of ty is RHS A337 toards BEALiue. (Rhinefield Ornamental Drive and Bolderwood Arbortereum Drive nearly join the areas.
If you're an early bird (6/7 am and check seasonal restrictions as you're banned in mid summer and possibly certain hours) you can ride the prom (along sea front at beach level) from Hengishtbury head (RHS of BMOUTH when facing map) to Poole – great kiddie cycle, BUT gets VERY busy after 9ish with dogs, pillocks, pensoiners, people who should no longer be wearing lycra etc etc but cafes are open etc..
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If you want a quiteish but stoney beach (little bits of sand) for a walk anbd a cafe overlooking IOW – Bbarton on Sea – lhsish