"My guess is that these [Zimmermans] are probably from the same town in Switzerland, probably emigrated together, and would be found together in Switzerland at some point ... How would I go about making the connection from the Zimmermans in this town in Ohio."
- Zimmerman is a reasonably common name in Switzerland/keep careful records.
In 1998, one dschmutz at es dot com provided some present day statistics. "Zimmerman" was twice as common as say "Arnold" but half as common as say "Schmid." All were less common than Müller (it led the pack); at a distance in second place was Meier.
According to Wikipedia, Zimmerman is derived from Zimmermann, a German last name meaning "carpenter." (Which might explain why it is so common.)
- Ohio towns and counties can be confusing/keep careful records.
There is a town, Tuscarawas, in Stark County, Ohio. I checked the 1850 U.S. census indexing at Ancestry.com. There were not any Zimmermans indexed in that census at the town, Tuscarawas, Stark County, Ohio.
There is also a Tuscarawas County in Ohio. There were 90 Zimmermans indexed at Tuscarawas County in that 1850 census. Of these indexed Zimmermans, 47 were born at Ohio (1809-1840); seven (7) at Pennsylvania (1809-1929); nine (9), at Germany (1813-1830). Twenty-seven (27) Zimmermans in the index (Tuscarawas County, Ohio) were reported born at Switzerland (1791-1840).
In 1850, those 27 Zimmermans (indexed at Tuscarawas County and born Switzerland) lived at different towns, several follow: (a) Auburn, Tuscarawas County, Ohio = 19 souls born Switzerland (1791-1840); (b) Bucks, Tuscarawas County, Ohio = 7 souls born Switzerland (1795-1840); (c) York, Tuscarawas County, Ohio = 1 soul born Switzerland (1830)
- Eat an elephant one bite at a time.
In many respects, you are asking the question, "How do I learn my [insert surname] family history?" It would probably be easier to count those not interested than to tally genealogists working day in and day out to identify immigrant ancestors, trace them to an earlier homeland and connect them to families here and abroad. Again, many/most of us find families with similar surnames living in close proximity to theirs and hypothesize about possible relationships.
I make better progress when I set out to solve specific genealogically relevant questions; develop a plan, working from the known to the unknown.
Genealogically relevant questions are specific: Who was the father of XXX died XXX, married XXX at XXXX on XXXX? [See Tom Jones, Inferential Genealogy.]
For each genealogical question, you'll want to develop a plan. The plan will depend on the work you have already done, what you have already documented. [See Kimberly Powell's "Think like a detective - Developing a Genealogical Research Plan."]
A good plan makes use of tried and true research methodologies, like the Genealogical Proof Standard, taking full advantage of technology.
A good plan will allow you to systematically locate and learn from a variety of different records. This means you want to learn about availability and accessibility of published histories (local and family) and different record groups, especially local records. (See the Genealogy.se question, "Help tracing ancestor back to [G]ermany?)
A good plan will make sure you interview family members and help you identify skilled researchers with an interested in these families. This work provides clues and possibly even a road map for your plan. In some cases, you may develop the opportunity to collaborate. In your case, consider corresponding with individuals listed below.
Professional genealogist, Fredric Z[immerman] "Rick" Saunders, Salt Lake City, who has traced some Zimmerman families from Tuscarawas Co., OH to Switzerland. His WorldConnect file is "Ancestry." See his professional page at GenealogyPro.com, "Fredric Z. Saunders." See also, "Fredric Zimmerman Saunders."
Kathlena, whose pages, "Tuscarawas County Ohio Family Roots" are part of WorldConnect. The file contains some information on one or more Zimmerman immigrants to Tuscarawas County, Ohio.
P.S. JustinY has provided you a general outline. In my experience, 19th century vital records and census at Ohio are must haves, but they do not provide much identifying information. Naturalization records of the day are again must haves, but often provide very little information. In my Ohio work, the more valuable resources are the extant probate, other court, land/deed, cemetery and baptismal records. The published area histories, especially for town and church development are also important. I have found local areas often have some set of unique and amazing collections--you learn about these by communicating with local societies and libraries.