Use redirection, for example:
ping 192.168.1.1 -t > filename.txt
This will redirect all (standard) output from the program in to filename.txt, which will be created if it doesn't exist and overwritten if it does.
You can instead use >> instead of > to redirect the output to a file and append the results to the end of the file, instead of overwriting (with thanks to @Jane T for the reminder).
Note that you will not recieve the normal on-screen output if you do this.
Update in response to comment
To delay between pings and record the time of each you can do some scripting.
Here is a quick Windows batch file I've thrown together, it prints the time, pings google, then waits for 3 seconds before repeating itself. I'm not a batch file expert so if anyone spots any problems please flag them up! And this probably isn't the "best" way to achieve what you are after - that might make for a seperate question really.
@ECHO OFF
:LOOPSTART
time /T
ping www.google.com -n 4
sleep -m 3000
GOTO LOOPSTART
Save this in a .bat file somewhere, edit the ping target and delay time as you need it, then run the .bat using redirection to pump the output of the whole thing to a file.
Note that this batch file never ends, but can be terminated by Ctrl + C and then Y if run from cmd. (You must press Y because it asks if you want to stop the batch file - even thought you cannot see the question because you've redirected the output!)