Both (Frehswater and Sea View) are beautiful spots, although SV is really crowded compared to when I was a child.
The brightly coloured sand comes from Alum Bay near the Needles (western end). You used to be able to scoop it up yourself from the beach, now I think you have to ge it from the souvenir shop, not recommended as it's the most ghastly sort of tourist trap (I should know - I was at school in Cheddar!).
If you go to Bembridge, at the Eastern extremity, it has a rather nice preserved windmill, either Eng. Herit. or NT. Some interesting woodwork in that.
Osborne House has some lovely tree specimens, as well as J.B.'s bench, and nice carving in the ballroom.
The church at Calbourne is an understated little gem (IMHO), and Winkle Street is worth a look - not 'woody but very sweet.
Non Woody:
If you get bored on a Thursday (check this), the Isle of Wight Military History museum, just outside Cowes on the road to Newport run their tanks on the rough ground behind the museum ( there's a good view from the high ground of the car park and you can ride in them for a fee). If the weather is dry it's spectacular, if you like that sort of thing (I do). Fantastic for kids. If there's a "gender split" on this, the shops in Cowes are twee/boutiquey, and it's close enough for her to drop you off first.
Again for kids mainly, Carisbroke Castle (Newport) has a donkey-wheel-driven well and good views too. Newport does get horribly clagged with traffic though. It's a real bottleneck.
Beken Photography in Cowes has been capturing the regatta for over 100 years, and you can get tinted prints. I love J-class yachts and they have many really good pictures.
The Needles batteries are worth a visit too. The old one has stunning views along the coast to Bournemouth (go early for less haze), and the new battery up on the hill was where the Blue Streak test stands were for the British Space Programme. We happened to see the Red Arrows displaying for Bournemouth Air Day from up there - probably a better view rthan the punters got.
Final woody thing:
Jason Heap, designer of amazing furniture, is based in Cowes, too, but I'm not sure if he has a showroom or if there are any local galleries showing his work.