The amount of available legislative history varies tremendously from state to state. What's more, getting this material can be tricky, frustrating and time consuming. So when someone asks me to compile a legislative history, I generally call a "Legislative Service" and hire them to do it for me. However, if I'm going to try to do it myself, the first step is to learn about the materials available particular state. Then I'll generally call the state library and ask them if they can get me copies of the materials I've identified.
Legislative Services: The Legislative Intent Service (800-666-1917) will compile legislative histories for any state that has legislative history materials. Other legislative services specialize in getting materials from a particular state. I've noted the ones I've used successfully in the entries for their states. To find other, you may want to post a query to the relevant AALL chapter listserv for the region. Note: While using a service will cost some money, it is could end up to be less than you would spend trying to get materials long distance. The service will also probably be more familiar with the procedures and materials available in each state and, hence, get better results.
Research Guides: There are a number of online legislative history research guides written for individual states. Most of these guides are listed at State Legislative History Research Guides on the Web. Another good bet would be the "Legislative History" chapter of the legal research treatises written for each state (New York Legal Research, Ohio Legal Research, etc.). You can probably get copies of the legislative history chapter from an academic library in that state.
I would also recommend (a) Lynn Hellebust's State Legislative Sourcebook (published by Government Research Service, 785-232-7720), which explains the types of legislative information available for each state and/or (b) calling a reference librarian in the relevant state library and/or (c) checking out the state library's Web site and catalog. For pre-statehood research questions, see Prestatehood Legal Materials: A Fifty-State Research Guide, Including New York City
And The District Of Columbia (Michael Chiorazzi & Marguerite Most eds., 2005).
Westlaw databases: Westlaw has notable databases containing legislative history materials for selected states, such as the State Legislative History - Historical (STATE-LEGHIST) and Governors' Veto Messages (STATE-LH-VETO). There are also individual databases for the states covered.
For more information on compiling a legislative history for a particular state, check the relevant Guide entry for that state.