My last garden was very shady most of the day, under trees, and shaded by buildings: here's some of what I grew both in and out of pots. I either used a tree and shrub compost (for trees and shrubs
) or mixed up a compost with half a good quality multipurpose like Jacks' magic and half John Innes number 2 compost with a bit of added grit and some slow release feed (Miracle grow does one , easily available, Wilkos usually has it) for perennials - the same but with Joh Innes Number 3. You can get all the compost ingredients easily at your local garden centre, Wilkos, B&Q and such. Local Nurseries usually seem to have pots for the best prices.
So, Plants ....
Ferns, hostas, epimediums, brunnera, lamium, certain grasses.
Acers work very well, mine used to live in the shadiest bit of the garden under trees.
Saxifrages like London Pride.
Tiarella, which is in flower at the moment and looks lovely. Also some of the Heucheras and heucherellas.
Aquilegia, Iris sibirica in lighter shade (thinking about it mine were in shade all year round)
Cyclamen, bluebells, anemones - esp the Japanese anemones (and pots will keep them contained so they don't romp over everything).
errm, what else did I have ... liriope, knautia, astrantia, some geraniums (I inherited a blue flowered one that was happy anywhere), aquilegia (granny's bonnets), geum, ajuga.
Anything with the second name 'sylvatica' means it comes from woodland originally.
Some of the Hardy fuchsias - I had the one with tiny white flowers that did very well.
Violets. Begonia and busy lizzy for summer colour.
Primula...
Most of these should be easy to find either at garden centres and nurseries, on ebay, or online from places like Long Acre Plants.
Oh yes, you could also try the smaller azalea and if you look at rose grower's websites you'll see some are suitable for shade - I had 'The Fairy' rose under trees.
I hope that gives you a few to get you started - Happy gardening!