Is the loud buzz noise the end of cycle buzzer, or is this a different noise? Sounds like the timer may have gone bad, possibly, not confirming that, not even safe to assume. Circuit half tripped I assume you mean this is a 240 Volt electric dryer and both breakers were found in the half way position. Breakers in the half way position mean something drew excess current and tripped them, breakers all the way over mean someone turned them off.
The buzz noise you hear may be normal, this could just be the timer running, you don't normally hear it because the drive motor and turning drum are too loud when its working. It could be a broken belt and the buzzing you're talking about could be the motor turning but with the broken belt the drum doesn't move.
As far as fixing it yourself it depends on your abilities and what tools you have, you'll need the basics like a screw driver and pliers, but more importantly you're going to need a voltage meter.
As far as paying $199 to come out and just look at it, ehhh, it's a tough call there, myself, I wouldn't. $199 but then you have to consider what parts it may need, sounds like a timer or a motor maybe, that could add another $100 to that bill. Brand new, or used dryers are cheaper than that, so how old is the dryer and how much did you pay for it new?
Dryers are about the absolute simplest things to trouble shoot and repair, but you would need at least some limited abilities to do so. If it were me, I'd stat by switching that breaker to the full off position, then back on just to be sure it reset fully. If this is an electric dryer you may be losing one side of power at the breaker, reset it to be sure. If its a gas dryer forget the breaker, if the light comes on and you hear noise you have power to it.
Unplug it from the wall before taking things apart, don't trust that you have the right breaker turned off, just unplug it. Plug it back in only long enough to do your tests, unplug it again as soon as your done. It likely won't kill you getting shocked as most people think, but it doesn't feel pleasant either.
Next step would be to remove the front of the dryer, being aware that the drum is held up in the front by the door area, so its going to drop when you pull the front off. Sounds like the belt isn't broke or you likely would of noticed while spinning it, but take a good look at the belt. CAREFULLY draw a picture on paper of how the belt is routed. Most people run into the most trouble just trying to figure out how to get the belt back on. Around the drum is obvious, but there is a tension pulley down by the motor, this is the routing area you want to pay the most attention to and the toughest part to get back together unless you've done several of them.
Pop the belt off, if you still have the door switch wires connected to the door close the door and try to run it, look in there at the motor and see if its turning or not. If it isn't you'll need to test for power at the motor, if its getting power but not turning then you need a new motor, if it is getting power and turning with the belt off then most likely the bearings are bad on the dryer and the drum is just binding up enough that the motor can't start, thus tripping the breaker and causing a hum noise.
No power to the motor then it sounds more like a possible bad timer, I'll assume the door switch is ok based on what you've said, and the start switch is ok since you do get a hum out of it.
Vacuum all that lint out of the bottom area before you put that part back together, keep your eye out for money, when I use to work on residential appliances I'd often find anywhere from $3-$5 just in loose change inside dryers.
Now you'll need to go into the top of it, most Kenmore's are made by Whirlpool and have a plastic strip on each far side of the control panel, carefully pop those off and you'll see a phillips head screw on each side going from the control panel side down into the top of the dryer, unscrew those, the front bottom of the control panel flips up and back. Plug it in, shut the door, set the timer, hit start, is this buzzing nose coming from the timer? Is the timer timing down? There's a little silver metal motor on the timer, its about as big around as a half dollar and maybe 3/4 inch thick, if you touch it is it hot?
If it sounds like you can do all of this yourself you'll likely be able to fix it and save yourself $200 plus, if not, consider just buying a new dryer. Unless this was a very expensive and fairly new dryer its not worth even $150 in total repair cost.
If you think you can do the above (very simplified directions) its likely worth repairing yourself, even if it needs a new timer or motor. It might not even need that much, but you'll know more once you open it up and have a meter capable fo doing electrical testing on it with. | 
Dryer will only buzz - not start 
