Hi, there are four sites on Anglesey that fit your requirements of Tents, showers, toilets, near beach, with a bar and shop on site, and you can view them here
You will also see the star ratings from visitors reviews on the link above to give you an idea which is the best of the bunch.
As long as you don't want five star luxury you could try Golden Sunset camping fields at Benllech, it's my favourite site on Anglesey and I'll be going on Friday for ten days. You don't book, you just turn up, but the site is so extensive you'll have no trouble finding a pitch, though it does get very busy at August bank holiday. Shower and toilet facilities on site and only a few minutes walk into the village with its shops, pubs and take-aways. A couple of minutes drive or ten minutes walk through the site next door will take you down to a great beach, and the Anglesey coastal path runs past the bottom end of the site. Brilliant views from most parts of the site. Also new for this season is an on-site catering van which opens for breakfast at 8am and stays open until 8pm at busy times. Cost is £7 per person in a tent, so it certainly doesn't break the bank
------------- Tigermouse
I have a very temperamental personality - 50% temper and 50% mental
If you are C&CC members (or would like to join on arrival) there are a couple of THSs scheduled for Tyn Rhos (listed in Ratty's link) this summer, as in previous years.
Also consider Penrhyn Bay, though I've not been there since before the earliest review!
Don't forget to leave a review of the campsites you have visited this year or last!