I heard we can connect Raspberry pi to laptop or PC Or Tv through LAN(Coaxial cable) and UART(serial to USB converter). Which one is gives better performance,speed(data rate). Instead of these two,is there any other mode of connection there? and also suggest me the power cable name?
The best connection you are going to get from a pi to anything is via USB, but that does not include normative peer-to-peer network connections (unless you get, e.g., a gigabit ethernet adapter). As Darth Vader points out, the USB and the ethernet jack share a bus, but the top speed of the USB side of things is much higher than the top speed of the ethernet side of things.
The ethernet jack is the second fastest and is capable of a normal, network connection. In theory most wifi adapters are potentially faster, but in practice this is unlikely to be the case. However, connection via wifi on a LAN can be roughly on par with connection via ethernet on a LAN.
A direct connection using an ethernet cable may be faster, but I do not think substantially so -- it may simply be more convenient, since you don't need a LAN, intermediate router, etc.
After two or threes years using B/B+/2B models, in practice I would say the USB tops out around 30 MB/s and the ethernet may do 10 MB/s in bursts but is more likely to sustain 6-8 on average, given no other bottlenecks.
UART(serial to USB converter)
You can reliably get 1 Mbps (i.e., 125 KB/s) duplex connections using the UART, but this is much slower than normal networking.