We just bought our house this weekend.
When the shower is used it starts to back up after about 5 minutes and the kitchen sink gurgles.
Last night I did a load of laundry. When I got up there was water on the floor of the laundry room and there was residue in the shower.
The laundry and the shower are in the back of the house the kitchen sink is in the front of the house. It appears the laundry drain is the farthest point in the house and I assume it feeds into the shower drain line and then to the kitchen drain and then to the street.
I also ran the dishwasher prior to the laundry. There was no gurgling and no water back up.

Did you have a home inspection done before you bought it? When I bought mine they turned on all the faucets for about 15 mintures to see if anything leaked or backed up.


But clogs can happen rather quickly.

Unfortunately I did not.
During one of the appraisals low water pressure was discovered. Because of that the seller did "all new plumbing."
I don't know for sure but I assume it would be supply lines but not sewer lines.

Yes... sewer lines are almost never replaced unless there are major issues with them...

Knowing we can't be sure when this started or how long its been occurring... my personal recopmendation (so you know what you are working with) if have the drains snaked out to the street... and have them run a camera in the pipes out to the street as well.

If you have collapsed pipes or other major problems (other than a simple debris type blockage) you might be able to argue this was an undisclosed defect.

And if they find no other problems you will have a degree of peace of mind.

I'm guessing you might find other problems in the house since you never got the inspection that might have turned them up before you went to close.

Should I snake each line- the laundry drain, the shower drain, and the sink?

While they are there any extra charge would likely be minimal. Though the likely problem is well beyond those or the backup would happen almost immediately.