Couple words of advice. Forget you ever heard of the Lakefront Bike Trail, it's terrible. Also, our demographic loss is our cyclists gain, Cleve is a great town to ride if you know the best routes. Weekends are a breeze, weekdays between 9:30am and 3:00pm, or after 6:00 are very accommodating. This is long winded, but it's by far the most interesting and easiest ride through Cleve.
I'm assuming you're coming in on Rt. 6 through Sandusky. Great ride except just East of Vermillion, becomes a 4 lane, busy, 50mph route with no berm for about 5 miles. You can drop south to bypass, but it's circuitous. It's the worst stretch of road between Sandusky and Niagara Falls.
Follow Rt. 6 all the way into Rocky River. At that point Rt. 6 becomes Clifton Blvd. You can stay on Clifton or drop 1 block north to ride on Lake Rd. Lake is more scenic, Clifton is wider and straighter. Eventually Lake Rd. dead ends into Detroit, head East on Detroit. This stretch is pretty urban, but also fairly accommodating to cyclists. (Detroit and W. 65th is a little cultural district with some good food). Once in downtown you'll cross the Detroit-Superior bridge over the Cuyahoga River, you're now on Superior Ave. (Turn left on W.9th, or W.6th for plenty of other food choices.)
Superior Ave. will take you right through Public Square, or you can drop north to St. Clair or Lakeside for less traffic. Heading east on any of these, you'll get to E.9th St. If you head north less then 1/2 mile on E 9th, you'll end up at the R&R Hall of Fame, Science Center and Cleveland Stadium, there's also a nice park at the very end of the street. Worth the detour.
Heading back east on St. Clair or Superior (incredibly wide outside lanes), you eventually want to make your way south to Euclid Ave. You can use E.18th, (you're now at my residence BTW), E30th or E40th for direct connections between Superior and Euclid. Heading east, Euclid Ave. has a great bike lane for the next 100 blocks (if you're going to have a heart attack, plan for it here, it's almost entirely Cleveland Clinic property
Okay, grab your map, this area was designed for effect, it's well worth it. Once you pass E.105th St. on Euclid Ave. you'll be in University Circle. Eventually you'll want Martin Luther King Jr. Drive heading north, but I'd suggest going a block further east and turning left on East Blvd. On your right is Severance Center (Cleveland Orchestra), on your left is Wade Lagoon (beautiful little inland lake). The area you're now in contains the fabulous Cleveland Art Museum, the Crawford Auto and Aviation Museum, Cleveland Botanical Gardens, Western Reserve Historical Museum, Natural History Museum and a few other interesting tidbits. Ride around, stop a bit, take a look around, it's just a great area of culture and interest.
Make your way back to Martin Luther King Jr. Dr. and head north. 4-5' wide bike lane the whole way. About 3 miles and you'll see I-90 freeway. Go past I-90 and MLK becomes Lake Shore Blvd (Rt. 283) through Bratenahl Ohio. If you want to see how old money in Cleveland is spent, here's your chance. Rt. 283 heads east past Mentor and Painesville, I rarely ride that way so can't really give you detail. Mentor is pretty congested so I'd look for side roads if that's your route.
That's a lot of words, but it's really a simple route. If you're interested, PM me and if I'm around we can meet up and I'll give you the tour. Either way, have fun.