Encourage them to read, join your local library, get them doing one or two organised things after school (don't overload them, they need their downtime), take them on interesting day trips, try and get them to watch some worthwhile stuff on TV or DVD at least occasionally, above all talk to them and set them a good example by reading and watching some quality stuff yourself. If there is someone in the family who plays chess, that's great as a way of stretching a bright child outside school.
My children went to a state primary school that many on Mumsnet would not touch with a bargepole. Ofsted reports not glowing, SATS results middle of the list in an inner London borough. My son was offered scholarships to three academically selective independent schools with no tutoring. (He might have been offered a fourth if he hadn't said at the interview that he wanted to go to one of the other schools.)
We went through some papers with him so he was familiar with the format and to check that he had covered everything they were likely to ask about. He had. They used setting for Maths from quite early on in the Juniors. He was in the top set and by his teacher's estimation by the time he left he was working at Level 6. They had a creative writing group for the most able children in his year to give them some extension work, and also as recognition that they didn't need the extensive SATS cramming that the other children were getting. He got Level 5s across the board in his KS2 tests and he wasn't alone by any means. Also it was through the school chess club that he developed an outside interest that really stretched him.
He had no trouble whatsoever coping with the transition to secondary school. The foundation he had from primary school was excellent.
Trust your local schools and your teachers!