There are quite a few connections throughout the network:
Long post:
LONDON UNDERGROUND CONNECTIONS
Piccadilly Line
The Piccadilly is the best connected of the tube lines. There are connections to the Victoria and Northern Lines. Between Barons Court and Hammersmith, District Line trains can transfer to the Piccadilly Line and vice versa. Finally, the line shares tracks with the District from Acton Town to Hangar Lane Junction, and with the Metropolitan from Rayners Lane Junction to Uxbridge.
Bakerloo Line
Southbound Jubilee trains can switch to the Bakerloo at the south end of Baker Street; northbound Bakerloo trains can switch to the Jubilee at the north end.
Central Line
Trains in the District Line platforms at Ealing Broadway can run on to the eastbound Central Line; this connection is normally padlocked.
Trains can leave Ruislip Depot at the west end and run along a single unelectrified line around the back of the station to a National Rail siding, and then back on to the northbound line to High Wycombe. There is also a single-track connection from the west end to Ruislip siding on the Metropolitan Line.
Circle Line
Almost all of the Circle Line is shared with either the Hammersmith & City Line or the District Line, and a significant portion with the Metropolitan Line. There are thus connections to other portions of each of these lines; all these are flat junctions:
Aldgate Junction Hammersmith & City Line
Baker Street Junction Metropolitan Line
Praed Street Junction Hammersmith & City Line
High Street Kensington District Line
Gloucester Road District Line
Minories Junction District Line
District Line
There is a connection to the National Rail branch from Romford at the east end of Upminster station.
Hammersmith & City Line trains run on the line from Barking to Aldgate East, where there is a flat junction. There is also a connection to the East London Line at St. Mary's Junction.
Circle Line trains run on the line from Minories Junction to Gloucester Road (both flat junctions; see the Circle Line for more details). The section from Edgware Road to High Street Kensington is also shared with the Circle Line.
There is a connection to National Rail at East Putney, allowing trains from Clapham Junction to run southwards on the line. At Wimbledon, there are connections to Wimbledon Park Depot (not LU); trains can enter from either end. In addition, at Wimbledon, eastbound trains can run from the up slow NR line on to the eastbound District, and from the westbound District on to the down fast NR line.
Piccadilly Line trains can run on the District Line tracks, and vice versa, between Hammersmith and Acton Town. Both lines share tracks from there to Hangar Lane Junction. At Acton Town, trains can also leave Acton Works (south of the line east of the station) into the westbound platform, and westbound trains (either Piccadilly, District, or trains having used the turnback sidings east of the station) can enter a shunting neck and then reverse into the works.
There is a connection to National Rail at Gunnersbury, and trains may run from the southernmost platform at Richmond on to the eastbound NR line to Clapham Junction. Finally, trains departing Ealing Broadway may run on to the eastbound Central Line.
East London Line
A double track link connects the line to the District Line just east of St. Mary's station (now closed); eastbound District trains can run over the link to become southbound East London Line trains south of Whitechapel; similarly northbound ELL trains can become westbound District trains.
There are no longer any links to the main lines at any of Shoreditch, New Cross, or New Cross Gate, though the southern extension will restore the last of these and the northern extension will provide a new link.
Hammersmith and City Line
The line shares track with the Circle and Metropolitan Lines between Aldgate and Praed Street Junction, and with the District Line from Barking to Aldgate East Junction; there is a connection to the East London Line on the latter section.
Jubilee Line
Northbound Bakerloo Line trains can become westbound Jubilee Line trains after leaving Baker Street station; similarly eastbound Jubilee Line trains can become southbound Bakerloo Line trains after leaving the same station. Metropolitan Line trains can use the Jubilee tracks between Finchley Road (exclusive) and Neasden (inclusive), and Jubilee trains the Metropolitan tracks between Neasden and Wembley Park (both exclusive)
Metropolitan Line
The Metropolitan Line runs over the Hammersmith & City tracks from Baker Street Junction eastwards, and shares tracks with the Piccadilly Line from Rayners Lane Junction to Uxbridge; the latter section also connects with Ruislip Depot on the Central Line. Metropolitan Line trains can also use the Jubilee Line tracks between Neasden and just north of Finchley Road (but are normally barred because of ATO issues).
The line also connects to National Rail at Harrow-on-the-Hill and Mantles Wood Junction.
Northern Line
The only way off the line is via the so-called "King's Cross Loop" - a single-track link between the Northern and Piccadilly Lines. A train in the northbound Piccadilly Line platfrom at King's Cross St. Pancras can run southwards on to the loop and thence into the northbound platform at Euston (the reverse movement, while physically possible, is not signalled). There is also a third track (the so-called "Euston Loop") from the end of the southbound platform at Euston to Euston Loop Junction (which is in the northbound track only) and thence on to the King's Cross Loop, from which the train can run north on the Piccadilly or reverse from the platform over a crossover to the southbound track. The Euston Loop, of course, is the old northbound track before the reconstruction at Euston to accommodate the Victoria Line.

click to enlarge.
Victoria Line
A southbound Victoria Line train can become a southbound Piccadilly Line train at the south end of Finsbury Park station; a northbound Piccadilly train can become a northbound Victoria Line train, also at the south end.