It sounds like you are using the meter incorrectly...
You CANNOT check, using the ohms scale, continuity with power applied. Doing so will either ruin the meter, or if you are lucky, just blow the fuse. On the continuity scale, 1(max) is open, zero is a direct short.
You have a serious problem that under the right circumstances, could kill you or someone in the area.
It sounds to me like your neutral is open. This would likely burn out the oven things like the lamp and timer, etc. The burners don't need the neutral, nor does the oven, so they could continue to work.
On new stoves, they are designed with a 4-wire plug. Two hot wires, neutral, and safety ground. In the stove, neutral and ground are potentially bonded together in a 4-wire circuit (depending on age of the stove), and must be if you feed it with a 3-wire plug.
Check the screws where the neutral comes in...both in the recepticle in the wall and at the plug and stove.
Since you don't appear to be very familiar with the use of a meter, you should probably call a pro. Do all of your checking with the fuse or circuit breaker off unless you are checking for voltage, where, obviously, it must be on.
I highly suggest you get a pro to fix this before you get hurt.