The best way to travel around Colchester is by bus, you can buy a day ticket for around £3.40 and it will get you most places in the town. If you are staying in the centre everything is within walking distance.
Colchester bus station is 1 of the embarassments of public transport in UK - giving foreign visitors the impression they've arrived in a 3rd world banana republic. NO digital timetable - no sector map to inform as to what service leaves from which bay - inadequate shelters over bays & no waiting room, so on rainy days, passengers just get soaked. Next door is a giant warehouse the local council squandered millions of taxpayers money on in the name of art, when this public cash should've been spent on building a bus station suited to the 21st century. For a destination actively campaigning to be raised to 'city status', Colchester bus station on East Hill is nothing other than PATHETIC...
Colchester has two stations. Colchester itself is a major station on the line from London that serves Ipswich, Harwich and Norwich. Colchester Town is a small station tucked away on the other side of the town centre. It is much closer to the centre of Colchester, but you usually have to change at Colchester and there are only two trains an hour. I prefer to use the main station - its not far from the centre, only a mile from the High Street - a short bus or taxi ride, but not too far to walk, although it is up quite a steep hill !
The services are operated by One Railway, the company that was recently formed by the amalgamation of Anglia Railways, Great Eastern Railways, Stansted Express and the part of WAGN (West Anglia Great Northern) that served the western part of East Anglia.
Colchester has a very good service from London Liverpool Street, with five off-peak trains an hour. Journey times vary from as little as 49 minutes for the fastest trains to one hour for the slowest, stopping trains. Timetables can be downloaded from the website in pdf format. The website has a fares section which describes the various types of ticket available, but unfortunately does NOT give the ticket prices.
For prices on-line you should go to The Trainline where you can put in your start and end stations and travelling time, and you will get the best ticket price. If you want to you can order tickets on-line, but it is better to buy at the station unless you need to reserve seats.
Colchester's main line railway station is a little over a mile out of town so either a bus or taxi is required to get into the centre of town unless you have no luggage and are fit enough to pace it out into the High Street up a pretty steep North Hill.