Send all the paperwork into the solicitor, it'll save them doing a letter to ask you to bring them. Solicitors do everything by post, not email, which takes longer. Let them know when you would like to complete by, and then phone weekly for a progress report. Phone them asap to check that they've ordered the searches, and they might want a cheque from you to cover the cost of these.
If you're buying and selling through the same estate agent that'll be easier, as they'll know what's going on with both and you won't get mixed messages. Tell them when you want to complete too, and phone them at least weekly for an update and to ensure they are dealing promptly with any issues that arise - they are your link with your buyers and the sellers of the house you want, so keep in the loop so that any problems can be sorted at an early stage.
Have you got your mortgage in place? You need to be getting that organised as soon as possible, as the mortgage company may want to do a valuation, and all this takes time, so speak to your bank or broker and get that moving too.
You can complete in less than eight weeks if everyone is really motivated and no problems arise, and the money is straightforward, but it's pretty unusual. But the more you get all your bits in place, and keep in touch with your mortgage company, solicitor and estate agent, the more straightforward it is likely to be.