Hi,
Before going to vlan topic I have basic question. I have a gateway as 10.209.108.1/22 if I am connecting gateway to cisco 3750 switch port 1 can I use remaining ports to connect via above gateway? I mean suppose I am connecting NIC to port 2 and IP 10.209.111.200 can i ping 10.209.108.1? Is this valid without configuring any vlan or anything on switch?
I am not able to ping gateway, what could be possible reason?

hard to know what you mean but 10.209.108.1 and 10.209.111.200 are on the same /22 so yes they should be reachable without any L3 device inbetween.

I have IP range of network 10.209.108.0/22 and I want to set up a VLAN to do this I have cisco 3750 switch. I had connected all my machines to 3750 switch and they can communicate (as expected). Now I want to connect this setup to internet, for that I connected cable from 10.209.108.1 switch to my cisco 3750 switch. when I tried to ping 108.1 gateway its Not working.
what I observed is after connecting gateway switch cable to cisco switch after some time that link becomes dead. these two switches has some problem to work with together. may be I am doing something wrong.
what could be possible reason?

i've no idea why it's not working, you've not provided the IOS config and such, so there's no way to know how the switch ports are set up. It may help to put an IP on the switch on that vlan and ping back and forth between each machine and the switch instead of going through the switch to other endpoints.