Garbage disposal&kitchen sink drain go to basement, directly below is my laundry room.
P-connection from washer to same drain pipe. When I run my washer, water is coming back out of the overflow drain to floor drain.

I pulled grill off floor drain, there is mucky water in this floor drain, standing in the hole.
There is also a plug with hole that comes out of the side of drain hole.

If I rent a snake where is the clog?

I recently put cucumbers down my garbage disposal, and I have seeds and onion bits standing in floor drain hole, not filling up the floor drain itself.

No drain cleaner has been used up to this point.

Can I fix this myself today?

You are clogged downstream from the floor drain. You have options, 1) you
Can snake from the floor drain,( you will have to work the snake around the return bend of the trap.)**OR** 2) You can snake from the washer or kicthen roof vent,(washer vent's closer to the clog).
If snaking from the roof put out enough cable to reach the base and 25 feet more. Good luck, Tom