Vibrations in frame when pedaling
Well, it's hard to say from your description.  Is the vibration in time with the turning of the wheels, or much more rapid?

A vibration that's in time with the turning of the wheels may be a warped rim, a rim rubbing the brake pads, a tire that's not seated properly on the rim, etc.  (The fact that this occurred after changing the tire suggests that last possibility.)

A vibration that's much more rapid than the turning of the wheels is usually due to resonance in the frame.  You'll usually see this more often when going downhill at high speed -- the front wheel will suddenly begin vibrating violently, such that you fear losing control of the bike.  What's happening is that minor vibrations in the bike (or even just due to the road) found a resonance in the bike frame, and the vibration becomes self-reenforcing due to the way it causes the front wheel to twist.

I'd suggest first turning the bike over or hanging it somehow so you can spin the wheels freely.  Observe the wheels carefully, looking at the space between the rim and the brake pads.  Look for any change of distance as the wheel turns.  Some small amount (a millimeter or maybe two) is to be expected on a well-used bike, but any more is a problem.  Also, if the rim touches the brake pads, that's a problem.

Observe from the side and look to see whether the rim moves in and out (decreases or increases in diameter) in relation to the hub.  There should be no detectible motion in this direction. 

Next look at the tire while the wheel is spinning.  Look straight on so you can see both sides of the tire as it spins, and see if either side bulges out at any point.  Then look from the side to see if at any point the tire seems to increase or decrease in diameter.  Again, a one or two mm variation in these dimensions is probably OK, but no more.  

Also look at where the tire disappears behind the rim.  There is generally a ridge in the tire a few mm from the rim, and it should maintain a fairly constant distance from the rim edge as the tire rotates.

(Since your problem began after changing the tire, these tire checks are likely to reveal that the tire is not evenly installed on the rim.)

While you have the wheels off the ground, grasp each wheel near the rim and push back and forth sideways, feeling for any play in the bearings.  You should not feel any play.

Finally, set the bike back on the ground, grip the front brakes to lock the front wheel (leaving the rear brakes free), and push the bike forwards and backwards.  You should not be able to feel any play in the "headset" (the bearing where the fork turns).