So many variations to house buying depending on your circumstances .....
When your offer is accepted you instruct a solicitor. You then notify the EA who your solicitor is and a memorandum of sale is issued.
Your EA and solicitor will ask you if you are having a mortgage etc to buy the property. They may ask for proof of funds for a deposit etc or if you're a cash buyer.
Then you instruct your solicitor to carry out searches on the property you're buying. Your mortgage company will send out a surveyor to check the valuation is correct. You may wish to have a structural survey carried out, there are different types which are more detailed and more expensive, if you're buying a new build there's probably no need.
Your solicitor will then begin enquiries where they ask questions about the property you're buying.
Once all enquiries have been satisfied, searches are back they will request the money from your mortgage company.
They will then discuss with you when to exchange, this is when it becomes legally binding. Contracts are then exchanged between the different solicitors.
Completion can happen anytime after exchange, it could be the same day or a week, 2 weeks later - this is when the money moves up the chain. Once completion happens (you will get a phone call to tell you) you can then collect your keys and move in.
That's a very brief summary.... other posters please add anything I've missed.