Luckily South Euclid isn't that far from Cleveland proper so
you should be able to find either a Train (Called Rapid Transit)
or a Bus that goes near where you want.
Here's the RTA (Regional Transit Authority) site.
http://www.gcrta.org/
They handle all public transportation in the area.
If you are going Greyhound from SD they will drop you off in the
middle of downtown Cleveland. That's the only station in the area.
At that point you're about 9 miles East from South Euclid.
Traffic is relatively light in the area, unless it's a weekday between 5 and 7pm.
Then downtown traffic can get pretty bad in some areas.
But going East shouldn't be too bad from my experience.
If something happens and you need to grab a cab, walk directly
west of the station and there's a hotel right across 13th street
(you are behind the hotel at that point, walk thru the atrium/corridor
or walk to 12th street and you'll be in the front.
Embassy Suites combined with some condo thingy I believe.
Hotels are pretty much the best bet for getting a cab downtown.