The Surrey Iron Railway
The Surrey Iron Railway, extending from the Thames to Croydon, was the first public railway in England, authorised by an Act of Parliament in 1801. Horses provided the motive power to haul trains of wagons. A further act of 1803 authorised the Croydon, Martsham and Godstone Railway. The line to Martsham opened in 1805, similarly using horses, but the extension to Godstone was never constructed.
There's a more detailed account of this pioneer railway by Peter Mcgow here, on the Wandle Industrial Museum site.
A contemporary watercolour of the Surrey Iron Railway.
The Stockton and Darlington Railway
The Stockton and Darlington Railway opened in 1825 but, although steam locomotives were used for the goods trains, passenger trains remained horsedrawn until around 1834. There's a Wikipedia article here.
The Canterbury and Whitstable Railway
The Canterbury and Whitstable Railway was authorised by an Act in 1825 and started using its steam locomotive 'Invicta' for hauling goods trains in 1830, a few months before the Liverpool and Manchester Railway inaugurated its steam-hauled passenger services.
There's a 'Wikipedia' article on 'Invicta' here.
The Liverpool and Manchester Railway
The Liverpool and Manchester Railway opened in 1830. The first Parliamentary Bill to authorise the railway was rejected in 1825 but a revised Bill passed the following year. There's a Wikipedia article here.
The London and Southampton Railway
The London and Southampton Railway was promoted in 1831 and by 1838 ran from Nine Elms to Woking, reaching Southampton in 1840. As the system expanded, the name was changed to the London and South Western Railway. See the article Origins of the Southern Railway: Part 1 - L.S.W.R.
The Bodmin and Wadebridge Railway
The Bodmin and Wadebridge Railway was authorised by an Act in 1832 and opened in September 1834 with a single locomotive 'Camel', a single passenger coach and not many goods wagons. The railway struggled for funds and 'Camel' was not joined by the second locomotive, 'Elephant', until 1836. See the Wikipedia article Bodmin and Wadebridge Railway.
The London and Greenwich Railway
The London and Greenwich Railway was sanctioned in 1833 and starting running part of the line in 1836. By 1838, it was carrying passengers between its London terminus at Tooley Street (now London Bridge) and a temporary terminus at Greenwich. The final terminus in Greenwich opened the following year. The London & Croydon Railway (below) opened sharing the first 1.75 miles of the London and Greenwich on the approach to Tooley Street, as did the London & Brighton Railway (below). The South Eastern Railway approval also mandated sharing of the 1.75 miles from London Bridge.
The Grand Junction Railway
The Grand Junction Railway was authorised in 1833 and opened to passengers in 1837. It ran from a junction with the Liverpool and Manchester Railway at Earlstown to a terminus at Curzon Street, Birmingham, adjacent to the terminus of the London and Birmingham Railway. See the Wikipedia article here.
The London and Birmingham Railway
The London and Birmingham Railway encountered significant opposition from landowners along the route. The first Parliamentary Bill seeking to authorise the railway's construction was thrown-out in 1832, but a revised Bill in 1833 was passed. The railway was opened to passengers in 1838. There's a short article here.
The London and Croydon Railway
The London and Croydon Railway received consent in 1835 and by 1839 was running between its own station at London Bridge and West Croydon. To do this, it shared the first 1.75 miles of the route from London Bridge with the London and Greenwich. For a time, the railway used the Atmospheric System of propulsion.
The South Eastern Railway
The South Eastern Railway was initially sanctioned in 1836. Its final approval resulted in sharing the 1.75 miles of the London & Greenwich from London Bridge, then following the route of the London & Croydon and finally the route of the London and Brighton Railway as far as Purley. In 1864, it extended to Cannon Street and Charing Cross.
The London and Brighton Railway
The London and Brighton Railway, later renamed the London Brighton & South Coast Railway (LBSCR) was authorised by an Act in 1837 and started running in 1841, sharing the first 1.75 miles of the route from London Bridge with the London and Greenwich and then using the route of the London and Croydon Railway.
The trickle of new railways rapidly turned into a flood in the 1840s. Shareholders in some of the earliest railways earned large dividends, turning all railway shares into "hot stocks" encouraging investors and leading to the period of 'Railway Mania' when the most improbable routes were promoted and dubious operators like George Hudson (the so-called 'Railway King') made fortunes for themselves, if not their shareholders. Wikipedia discusses Railway Mania here and the notorious George Hudson here.
One of these 'Later Railways' was East Kent Railway, which I outline below because of its interactions with earlier lines. Dissatisfaction with the services provided by the South Eastern Railway in North Kent led to the bill for the East Kent Railway being introduced into parliament in 1853. In 1859, the railway was renamed the London Chatham and Dover Railway (LCDR). Although funding was a perpetual problem, the railway managed reach to Blackfriars and link up with the Metropolitan Railway before collapsing into bankruptcy. The LCDR somehow managed to carry on and, in a joint venture with the London Brighton & South Coast Railway, Great Western Railway and London & North Western Railway called the 'Victoria Station and Pimlico Railway' (VS&PR) participated in the construction of Victoria Station, first opened in 1860. A parallel station to the east, purely for the LCDR, opened in 1862. Faced with continuing financial difficulties, in 1899 The LCDR made an agreement with the South Eastern Railway to operate and market jointly, trading as the 'South Eastern and Chatham Railway'. However, the two railways remained legally distinct until 1923 when both, and the VS&PR, were absorbed into the Southern Railway by the Grouping.
 'Views on the Liverpool and Manchester Railway Taken on the Spot by Mr. A. B. Clayton', reprinted by Frank Graham 1970 (SBN 900409 29 0).
 'Views of the Most Interesting Scenery on the Line of the Liverpool and Manchester Railway', I. Shaw 1831, facsimile published by Hugh Broadbent 1980 (ISBN 0-904848-05-1).
 'British Railway History 1877 - 1947', Hamilton Ellis, published 1959 by George Allen and Unwin.
 'The Stockton and Darlington Railway 1825 - 1975', P. J. Holmes, published by First Avenue Publishing Company.
 'The Liverpool & Manchester Railway', R. H. G. Thomas, published 1980 by B. T. Batsford (ISBN 0 7134 0537 6.
 'The Railway Mania and Its Aftermath 1845 - 1852', Henry Grote Lewin 1936, reprinted 1968 by David and Charles (7153 4262 2).
 'Drake's Road Book of the Grand Junction Railway', 2nd edition 1838, reprinted 1974 by Moorland Publishing (ISBN 0 903485 25 7).
 'Britain's First Trunk Line - The Grand Junction Railway', Norman W. Webster, published 1972 by Adams & Dart (SBN 239 00105 2).
 'Oldest in the World - The Story of Liverpool Road Station, Manchester 1830 - 1980', edited by C. E. Makepeace, published by Liverpool Road Station Society/Manchester Region Industrial Archaeological Society (ISBN 0 907172 01 6).
 'History of the Southern Railway' by C. F. Dendy Marshall, revised by R. W. Kidner reprinted 1982 by Ian Allen (ISBN 0 7110 0059 X).
 'The London & South Western Railway' O.S. Nock, published by Ian Allen.
 'The South Western Railway' by Hamilton Ellis, published 1956 by George Allen and Unwin.
 'The London Brighton and South Coast Railway' by C. Hamilton Ellis, 1971 edition by Ian Allen (SBN 7110 0269 X).
 'The London to Brighton Line 1841 - 1977' by Adrian Gray, The Oakwood Press.
 'The South Eastern and Chatham Railway' by O. S. Nock, 1971 edition Ian Allen (SBN 7110 0268 1).
 'London's First Railway - The London & Greenwich' by R.H.G. Thomas, published by Batsford Paperbacks (ISBN 0 7134 5414 8).
 'The Railway Companion, Describing an Excursion along the Liverpool Line' by A Tourist, published 1833, reprinted by Deanprint Ltd. in 1980 and published on their behalf by Liverpool Road Station Society (ISBN 0 907172 00 8).
 ‘A Regional History of the Railways of Great Britain: Volume 2 Southern England’ by H. P. White, 4th edition published by David & Charles (ISBN: 0-7153-8365-5).
 ‘A Regional History of the Railways of Great Britain: Volume 10 The North West’ by G. O. Holt, revised Gordon Biddle published by David & Charles (ISBN: 0946537 34 8).
 ‘A Regional History of the Railways of Great Britain: Volume 11 North and Mid Wales’ by Peter E Baughan, published by David & Charles (ISBN 0-9153-7850-3).
 ‘A Regional History of the Railways of Great Britain: Volume 12 South Wales by D. S. M. Barrie, published by David & Charles (ISBN 0-7153-7970-4).
 ‘Atmospheric Railways – A Victorian Venture in Silent Speed’ by Charles Hadfield (David and Charles, 1967).