If I understand you correctly, you are quite happy with your current AD but just wish to upgrade it to a later version.
This is the smooth way as I see it when you have a second server standing by for commissioning. Note however that I am presenting a simplification and not in any way a complete guide (to which entire books are dedicated), there are quite a few caveats with every step which I am not covering. You will therefore have to make a bit of research yourself, but this is the outline as I see it.
Assuming you don't have applications with dependencies which would break in a newer AD version, or during a transition, you could promote your new computer as a second DC in your old domain thereby making them both serve clients during the transition.
When you are comfortable with the new DC in terms of it operating correctly and that DNS and backups are fully functional for the new DC, you can transfer the 5 FSMO roles from the old to the new server. Assuming DNS is accounted for, this makes the new DC self sufficient in the domain, and the old server a functionally redundant DC.
You should now be able to demote the old DC, meaning to uninstall the AD from it. This is a non-reversible maneuvre (discounting restore from backup). It is therefore usually a good idea to disconnect the server for a day or two before (reconnecting then) demoting just to see that everything works smoothly.
Again pay attention to DNS for both clients and servers and that backups are in order for the old and new server.
When the old DC is decommissioned and everything tested and still working, you can raise the forest and domain levels to get your new AD version.