Yes, you can do that without rooting.
Firstly, WiFi and mobile network are two totally different things. Android will always use WiFi first, and if it's not available it will use your mobile network for data.
Using WiFi isn't related to what SIM you have in the phone. You can use WiFi without any SIM in the phone at all. Your SIM card only affects whether you can use your mobile network for data or not.
So, in theory, all you need to do is turn WiFi on.
However, WiFi is unreliable, and the phone will not use WiFi by default when the phone is in standby (to save power). So, you will find that even with WiFi on you will end up using mobile data, which may or may not work. If it does work, it will probably cost a fortune without a data plan.
So... my advice would be to remove the SIM from your phone (this will ensure that it can only use WiFi), start the phone up, run the Market app, find APNDroid on the Market and install it (don't run it yet). Turn off the phone, put your SIM back in, turn the phone back on. When it's booted, run APNDroid to deactivate all of your APNs (APNs are your mobile data connections).
Once you've ran APNDroid with your SIM in the phone to deactivate your APNs, your phone will no longer be able to use the mobile network for data, and will only be able to use WiFi.
You can make your APNs work again at a later date if you get a data plan by simply running APNDroid again.
I'm not sure what you meant about using another SIM to make it work. You don't need to use any SIM for WiFi at all, but, if you mean the initial setup process that you need to do (entering a GMail account etc) when you start up the phone for the very first time, then yes, you can use someone else's SIM for that which has a data plan.