You have to keep in mind that there are an usual amount of suburbs in Cleveland. Not sure where you are coming from, but unlike a lot of cities most people in the metropolitan area live in the suburbs, which are all pretty small in terms of geography and population. Cuyahoga County alone has over fifty suburbs, so to go through all of them would be laborious, but picking the right suburb is important (because like I said there are so many and they are all a little different).
If your job is downtown Cleveland, then it probably makes sense to pick a suburb west of Cleveland with access to I-90. Your commute will be the shortest and most convenient. The suburbs along the lake or off I-90 are the nicest on the West Side. So this means suburbs like Bay Village, Westlake, Rocky River, and Lakewood. Bay Village is right on the lake, has a good school system, and isn't expensive as Rocky River. However Rocky River has the better school system older homes kept up very nicely, and a lot more character and nice restaurants and shops. Westlake is a lot of new construction, and you can get a lot of house for the money there compared to Rocky River. Westlake however has a lot of chain restaurants and retailers and its schools aren't quite as good as Rocky River. Westlake however has Crocker Park, which is one of Cleveland's best shopping centers and a nice place for families and teens to go. Lakewood is nice with beautiful older homes in neighborhoods along the lake. There are a lot of distinguished areas in Lakewood and the commute to downtown is probably only five minutes. It is safe, but the schools aren't that great and a lot of people send their children to private schools. Generally speaking the West Side is more new money and Catholic compared to the East Side. It is very popular to send children to parochial grade schools and a number of single sex catholic high schools on the west side. The west side suburban communities appear more midwest.
The East Side suburbs are best for commuting to jobs in University Circle (the city's second hub of employment and cultural center on the eastern edge of the city). The East Side suburbs are also the most affluent where median incomes for a community run from over $100K to over $200K. The nicest suburbs on the East Side include, Shaker Heights, Beachwood, Pepper Pike, Gates Mills, Hunting Valley, Chagrin Falls, and other small townships between these suburbs. These suburbs and the other better ones on the east side generally have access to I-271 which runs north-south on the eastern edge of Cuyahoga County. Shaker Heights is older homes, but is still probably the suburb with the most beautiful homes and very clean. For instance in Shaker Heights it is not allowed to have garbage cans on the curb waiting for trash collection. Garbage collection is taken directly from near the garage so as not to disturb the aestetic value of the city. The schools are also very good, and a lot of students end up at ivy-league colleges. Beachwood has newer homes than Shaker Heights and has an excellent school system and the region's most upscale shopping mall with Saks and Nordstroms called Beachwood Place. Communities like Pepper Pike, Gates Mills, and Hunting Valley are all filled with country estate types of homes. They have acreage requirements so lots typically cannot be any smaller than an acre. Chagrin Falls and Solon are other East Side suburbs that boast really nice homes and some of the area's best school systems. Chagrin Falls is probably has more of an attractive landscape and amenities, but Solon certainly has a lot of new construction. The problem with Solon and Chagrin Falls though, is that in my opinion the commute to employment centers would be a little too much. The east side suburbs compared to the west side are generally more old money and Jewish. Even though the schools are great, there is still the option to send children to very elite private schools. The east side really appears more East Coast than Midwest.
The neighboring communities in surrounding counties like Lorain, Lake, and Medina counties are seeing a lot of new construction. However, I would still be warry of considering these communities even though you could get a lot of house for the money and pay lower taxes. The schools really just aren't as good as the ones in Cuyahoga County, and the homes don't seem to be kept up quite as nicely after they are built.
The major realtors that serve Northeast Ohio are Realty One, and Howard Hanna Smythe Cramer. I would recommend browsing their websites for homes or contacting them to get an idea about where to live. Another insiders tip to picking a good suburb is to consider if the suburb has or is near a Heinen's Grocery Store. Heinen's is an excellent first class grocery store exclusive to Northeast Ohio and is located in the region's most affluent suburbs.
(By the way, I'm not from Rocky River, but that would be my first choice where to relocate).