What is the best way to switch between different ranges when designing a multimeter using microcontroller?
You don't give a number for the maximum delay you require, or much description of the design, but I'm assuming it's to be an autoranging/switching type of meter.

There are plenty of cheap analogue multplexers available, the best known being the 74xx4051, 4052 and 4053 series.
These are dual polarity 8:1 (4051), 2 x 4:1 (4052), and 4 x 2:1 (4053) muxes, available for under 50 cents and made by various manufacturers (e.g. TI, AD, Maxim, etc) I would have a look at these and similar offerings. Here is an example 4051 part from Farnell (many more here). Also see analogue switches.   

For slower switching only one or two ways, but better isolation and lower resistance there are mechanical relays (or solid state relays, such as the PhotoMOS type from Clare) These can be used where appropriate, together with IC switches, e.g. to isolate one channel from the next when active.