I recently moved into a home with a 3 prong outlet for my 4 prong Kenmore dryer. I went to Home Depot to replace the 4 prong plug with a 3 to match the outlet. I was ready to install the new plug when I noticed this white wire (2nd picture) and I'm not sure where it should be connected to. Originally on the 4 prong plug this white wire was connected to the middle screw (1st picture) but from the videos I've been able to see this seems to go to the ground (green screw). So my question is where should this white wire be connected to, the green screw or should I connect it once again to the middle bracket?
If you're going to a 3-wire cord (not recommended), you'll connect the white neutral/ground wire to the green ground screw.
The on the terminal block you'll have L1, neutral, and L2.
Here's a diagram from a random Kenmore electric dryer installation manual.
Here's another diagram from a random Kenmore electric dryer installation manual, this time showing a white bonding wire.
Getting power to the machine is the easy part. The neutral goes to the silver screw in the center, and the two hots go to the brass screws on either side. Now the machine is powered.
Safety ground is the hard part. You really want that.
If you have a 4-wire cable with NEMA 14-30, you're all set.
If you have a 3-wire cable with NEMA 10-30, you have several options, and the rest of this post is about that.
For dryers and stoves specifically, Code allows you to "cheat": bond ground to the neutral wire. The rationale is there are a lot of old NEMA 10-30's out there, and trouble is unlikely since dryers and stoves are rarely moved and plugs are rarely unplugged. The hazard is, if the neutral fails, it will energize the chassis of the dryer at 120V and will electrocute users, and has. Sometimes fatally.
Install a 4-wire cable to a 14-30. For a dryer circuit, Romex 10/3 is the right stuff, it sells for less than $1/foot. Follow the old wire and replace if feasible. If you're not comfortable going inside the service panel, leave about 4 feet of excess length and call an electrician to make the connections. If you prefer, you could run metallic or non-metallic conduit and use single-wire THWN wire (preferably stranded for easy pulling). Ground is optional in metallic conduit, the conduit is a legit ground path.
Replace the NEMA 10-30 outlet with a 14-30 by running a separate ground wire. It must be at least 10 AWG - single wire will do, but the wire must be bare, green or greeen/yellow, and re-marking another color is not allowed. Route however you can, to either the service panel inside on a ground lug, or to a ground screw inside a box in all-metal conduit that is continuous back to the service panel. This is covered under NEC 250.130(c). Don't connect anywhere else, e.g. not to 120V outlets, those have 12 or 14 gauge ground wires - too thin. You might be able to bond to a water heater's ground, but ask a code expert before you do that.
You cannot buy or hack a 240V-only dryer (which takes NEMA 6-30) and replace your plug and re-designate the white wire to be ground instead. Electrical code does not allow that. You can remove the insulation, but you must remove all of it - hard to do in jacketed Romex.
If you install any wire or grounds, get a book on wiring installation and learn how to do it properly and legally, so the wire is not damaged and any electrician who touches it won't want to tear it out and redo.
On a 4-wire cord , the middle screw is presumably where the neutral wire is connected.
On a 3 wire (and the whole reason 3 was replaced with 4) neutral and ground are the same. So white and green are connected, at least until you get around to running a 4-wire outlet. Normally detailed instructions are in the owners manual for your dryer - not infrequently they are also printed on the access plate for the wire connection, or a label nearby. If the "all gray" thing is your new cord, you probably need a jumper from the green screw to the center connection (and the white would go there as well.)