@zenbike is right on this - it is a common problem. What happened was that the old, 'stretched' chain damaged the rear sprockets. Look closely and you will see that the shape made by the sprockets is an elongated 'u' shape, particularly so on the sprockets prone to skipping (the smaller sprockets).
There is the option to replace the sprockets, however, you will need a special tool to remove the sprockets and you may not have the money for the tools + sprockets right now.
In the past, when confronted with this situation, I have continued to ride and, after not too long a time, the new chain 'stretches' to suit the sprockets and no longer skip. It need not stretch to the length of your old chain, just 'grow' the tolerances required to work with the setup that you have. It is up to you whether you follow this 'cheapskate' option, however, you will need to remember not to ride out of the saddle on the gears where skipping is a problem.
Since you may have already damaged the new chain anyway and it will not be suited to a brand new sprocket set, you may want to see how you get on running 'cheapskate'. However, I cannot stress enough the importance: do remember to ride in the saddle as a jumping chain when out of the saddle is likely to cause you an accident when you least need it.
After a short period, surprisingly short, the tolerances should be good and you'll get a bit more life out of your sprocket set. Eventually though you will have to replace sprocket+chain. Don't wait for the gears to start jumping again, check the wear of the chain. Too much wear will damage the front chainset and that will cost real money.
There are tools for measuring chain wear, personally I prefer to just put a screwdriver under the most forward part of the chainset when the chain is on the outer ring and see how much you can pull it forward. If this gets to more than 5mm or so you know it is a badly worn chain. That is pure rule of thumb requiring experience, but that has worked for me in the past and I do not personally own a chain wear tool.