picturesque spots in North Wales

[Polar Bear]

Any suggestions please?   

[Wowbagger]

All of it?

Actually, no. Avoid the coast. Some of it's OK but there are too many people.

[Hot Flatus]

Shell island? Beddgelert?

[fred the great]

Port Merion (sp) based upon an Italian village I think. Both Snowdon and the Port Madoc Mountain railways plus of course many parts of Angelsey.

[Canardly]

Anglesey has some stunning views, is not overcrowded and has just spent a fortune on new cycleways. The Llyn peninsula is lovely but is very popular with Mancunian and Liverpudlian tourists particularly at week ends.

[Exit Stage Left]

It depends what you mean by picturesque. If you want some arresting images I reckon that the Dinorwic slate quarry has a lot to offer.
http://www.flickr.com/groups/dinorwic/

[redshift]

Cwm Idwal, and the Carneddau and Glyderau either side of  the A5 near Llyn Ogwen, if you like mountainous terrain.

The western edge of Holy Island, from Holyhead to Rhoscolyn if you like cliffs and tide races rather than long sandy beaches.

Rhosneigr to the Menai if you like long sandy beaches rather than cliffs.

Barclodiad y Gawres and Bryn Celli Ddu if you like burial chambers

Abergwyngregyn and Aber Falls if you like waterfalls.

Of course, quite a lot of this isn't approachable by bike.

[Efrogwr]

They are all right!
 It depends on the length of your trip. Anglesey is superb; it has an incredible variety of landscapes (because of its outrageously complicated geology) in a fairly compact area. Llyn is wonderful. Go to the far end (past an imaginary line from Nefyn to Pwllheli). Either of them would be great for a weekend, and leave you with the desire to explore further.
 If you have more time...
 Get the maps before you come! I much prefer the 1:25,000. It's a lot easier to relate the map to the landscape. However, 1:50,000 is more convenient on the road.