Hi,
I live about 4 miles from Pickering - and finding good, easy access fishing for trout on the local rivers is frustrating unless you're a member of a local club (and I've been on a waiting list for a couple of years now...:

)
From the list given on the Ryedale website, the Wykeham Lakes dayticket will do 'what is says on the tin' and is about a 20 minute car ride east along the A170. The Pickering Trout Lakes is well used by visitors to the area.
Don't know how the Dalby Beck fishes - the beck is a narrow, shallow water with trees on all sides, could be worth a look, but practise your sidecasting!
Closer to Pickering, the length of the River Seven at Marton on a £3.50 day ticket will get you 'on the water'. The length available is about half a mile. The river here is 15feet wide, max 4 feet deep and is mixed with trout, grayling, roach, dace, chub etc. You can get a good afternoon / evening out of this stretch - and there is a good hatch of mayfly coming off at the moment.
Finally, in the village of Sinnington there is about a quarter of a mile of free fishing on the Seven from the bridge in the middle of the village downstream - similar to the Marton dayticket water.
The contact listed at the bottom of the Ryedale list - Roger Beck - is a top guy and lives about 20 miles from Pickering, he might be worth a direct e-mail.
Good luck with whatever you manage to work out.