Ok--
(I'm a diy-er, not a plumber)
At the meter side, it looks like that funny coupling doodad is threaded onto brass. I would remove that and use a "female copper adapter", which allows the transition from threaded pipe to sweated copper.
At the end point, it looks like a brass coupling, and I would either keep the coupling and put a "male copper adapter", or remove the coupling and put a female copper adapter on the pipe.
Here's how:
I would take 1-2 feet of copper pipe, sweat on the adapter, and THEN dope up the threaded end (when cool, of course), and thread on to the existing pipe. This way, heat won't mess with the teflon tape/pipe dope used on pipe threads.
Then, sweat on a copper coupling on the copper tube, and continue the run.
If you need to bend the pipe slightly, you can use a "hickey" used for electrical tubing. Or your knee. :)
People disagree on what to use on pipe threads for sealing. I've had inconsistent results with teflon tape. Old hercules pro dope and thread is a pita, but guarownteed. They make new high falutin and pretty pricey teflon pipe thread dopes, that plumbing houses carry/recommend. They seem pretty good.
You gotta know how to sweat copper. Some people have a knack, others need some practice. It's real quick, once you know how. Clean super brite w/ coarse steel wool, flux, sweat.
Not kidding about the garden hose. They make fittings for garden hose, so you can make temporary connections until you finish the job. ie, sweat so the joints don't leak. :)
When using a pipe wrench, always use two, so that you don't stress the pipe "downstream". Breaking inaccessible joints can be a bitch.