What paperwork do I need to get the immigration process started?

There are many ways to file, and it depends on where he is and where you want to file. Most cases start with your i-130 relative petition; then if he is in the US, he files the I-485 application for adjustment of status to permanent resident or, if he is outside the US, either the K-3 visa for a spouse of a US citizen to enter the US and file the adjustment of status or the immigrant visa directly. Consult with an immigration attorney.

You can consular process him by initiating an I-130 petition. This process should take about 9 to 12 months.

Looking at your question, I do not recommend you do it yourself [!]. Hire an immigration attorney. I have seen a lot of people that rely on Internet information and think it is easy enough. BUT I see a lot of them come to see me months if not years later frustrated and ask for my help.

There are several factors relevant to what process you take. First, is your husband in the US or in Jamaica? If he is in Jamaica, you have to decide whether to file through the Jamaican Consulate Office or whether you want to file for a K-3 visa. If your husband is in the US, and if he qualifies to adjust status in the US, you need to file an adjustment of status. A standard adjustment of status usually takes between 4 and 6 months, but the specifics of a case may cause it to take longer. If you are unsure of how to go about the process, I recommend obtaining the assistance of an immigration attorney.