This varied from place to place, government to government, but for the most part was the same basic process. Normally you have to fill out a form, pay a fee, submit your request, and wait. I had to request vital record documents from Arizona, Nevada, and Pennsylvania. Obviously I also had to order documents from the USCIS and the commune in Italy.
By far, the most user friendly experience I had was Clark County's website for finding marriage records. They have a form that you can do your search right there on the site, and if your record is found, you can order it right there. This was the easiest, so if you have some docs in the Las Vegas area, you're in for a treat! If not, get some envelopes.
The system that I went with was a bit expensive on shipping costs, but one of the most important things I did was figure out and organize all of my information ahead of time. I saved hundreds on shipping costs by combining requests in the same envelope. The government agencies were happy to use my one return envelope as well. Seems elementary now that I'm typing it, but I didn't think about it until well into getting documents.
I was doing a lot of this mailing during stupid covid so I wanted to figure out how I could do as much as I could from home. Not completely associated with dual citizenship, I decided to get a laser printer. I believe this cut down on printing and copying costs. I was weary about making the jump from inkjet to laser, but I'm glad I did and that's why you're reading about it now. Here's the link to a Brother that looks the same, the toner you can get cheap on Amazon as well. We've had ours for at least 3 years, use it frequently, and have only changed toner once.
When I started, I went to the post office and got some priority mail document envelopes. While I was there I bought some rigid cardstock 9" x 11.5" mailers. The priority envelopes are free, and the cardstock ones I think were about a buck each. When I ran out of priority mail envelopes, I found that the cardboard mailers fit inside standard size manilla envelopes as well. I had a lot of those sitting around so that did save me another trip to the post office. Now that covid's different (not counted as much and everyone has it) going to the post office probably isn't as much of a concern. I did choose USPS specifically because you can print the labels at home, and they will pick up right from your house (in the US).