I replaced the rear cog as the chain I had snapped and I thought it had taken some teeth off. Now when I pull away in 1-3 gears it's ok, but if I am in gears 4-7 and I try to pull away ( say I have stopped at lights and not had chance to change back up to 1st) then the chain slips and bounces off the cog its on. If I change up through the gears when I am already riding it isn't a problem, just seems to happen if I put any tension on the chain in the higher gears when I have stopped and start pulling away again. Could this be a cog problem? set up or chain problem? Its done it for a while now and I have put up with it, but I am overhauling the bike and would like to have a heads up as to what I should be looking at to fix/ replace.
When you replace the cluster (rear cogs) you almost always need to replace the chain too because they wear together. If you put an old chain on a new cluster, you often experience that kind of because the length* of the links in the chain is slightly longer than the spaces between the teeth on a new cluster.
*Note the links don't actually get longer, it's just that the rollers wear out and allow a little play and the links sit a little further apart. This is known as "stretch" though it's different to the way something like elastic stretches.
Look at the chain closely as it passes over the cassette. It should be even, not with gaps under the chain before/after the contact point.
I'm now wondering if you've got a mismatch between the chain and cassette. Is the chain a 6-8 speed one of the same width as the old chain?
edited because I didn't read the initial question properly