You have a lot of work to do in the USA, Travis, before jumping to resources in Germany; and in accumulating information in the USA, and reading this list faithfully, you will find many clues to your Hanover heritage.
You are very fortunate to be researching an Illinois family. You have wonderful resources close at hand!
Online there is a statewide marriage index on the secretary of state web site that will give you all the information you need to send for copies of all your ancestors' marriage certificates which, depending on time and place, may give parents' names, places of birth for bride and groom, occupations, previous marriages, etc. Also check for and send for their siblings' certificates, which sometimes give you a lot more and better information -- again depending on the time and place and person taking down the information.
Online there is also an index of Illinois deaths from the state archives, which will also give you all the information you need to start collecting your ancestors' and their siblings' death certificates -- and the vast wealth of information those contain.
Illinois also has regional archives depositories -- state records that are stored at the regional state university libraries such as Northeastern Illinois University in Chicago, Illinois State University at Bloomington-Normal, etc. These regional archive repositories store records for the particular region in which they are located -- so that, for instance, if you live in Chicago but you are researching ancestors in Bloomington, you would go to that web site and then call, email or visit the regional depository in Bloomington at Illinois State University.
Their collections vary to some extent depending upon which records exist in each region. In Cook County, for instance, the only 19th century voter registrations that exist are for 1889-1891 in the City of Chicago. Another regional repository might have voter registrations from 1850 or earlier, but perhaps only for one or two of the five or six counties in that region.
So you have to do footwork to reach your own family in their own time.
Use the mailing lists at rootsweb.com and the genealogical and historical web sites for the cities and counties in Illinois where you are searching. Read and ask your questions there. They are often wonderful resources -- as well as occasionally turning up another researcher who will share information and leads with you!
Church records are also a good suggestion. Once you know where they lived at various times and when and where they were buried, you can go to www.familysearch.org and seek out the online catalog to look for microfilms of all their religious records -- baptisms, confirmations, marriages, etc. from the churches you know that or guess that they attended from all the information you have gathered on the marriage and death certificates.
Often the church records have more information than the civil records, and just as often they do not. Again, in records, all depends on the time and the place and the care taken by the officials involved in recording the events.
And beware: those who were Lutheran in Germany may have attended a Baptist or Methodist church in Illinois -- or no church at all -- if there was no Lutheran church for 20 miles around!
At the family search web site you will look for a list of FAMILY HISTORY CENTERS in order to find the location nearest to your home where you can access those microfilms of the church records you want. Go there with all the information you can copy from the online listing for each church you seek. They may have some or all of your films on hand at your center. Or you can request that they rent those films for you, one or two at a time, for a very nominal fee.
Swiss, Germans and Scandinavians often came with "references" from their pastors at home and these are sometimes recorded on the microfilms, also.
Good luck!
Maureen