My company was acquired by another company. Both companies have existing AD forests and domains. The company that acquired us ,also has international sites.What would be the best practice/solution to get both AD forest and domains working together?
Back up everything! AD is one complex and picky beast. You don't want to be responsible for bringing down a whole domain.
If your using exchange the first step is usually too change the mail domain for all of the users of the company which was purchased to the head companies domain. You will have to set up shared address space SMTP and Mail routing for this to work.
Afterward you will want to create a trust between both forests to allow users to interact between forests. There are a lot of options here to determine how the trust operates and every organization does this a bit differently. I suggest reading up on it a lot before you attempt it.
Technically speaking that's all you have to do to make it work, but it's not very clean. I would probably use a tool like ADMT to actually merge the forests after I had everything working. But keep in mind that there are lots of small things that can break this. Namely SID checking and a few other small bits. This is where a real AD expert would be handy. I understand that's not always possible, the only reason I learned to do this was because I found myself in your same position and the higher ups looked to me to make it happen.