As for the M77 versus the M77 Mk II...
Ruger used contractors for barrels during most of the years that the original, tang safety, M77 was in production. They generally stuck with Douglas barrels from '67-'73, and Wilson barrels from '73-'91, with other contractors filling in when demand was high. So, performance was all over the map. Wilson barrels, in particular, earned a reputation for printing shotgun patterns, and being insanely terrible when warm.
The M77 Mk II, however, has used Ruger's own (in-house) hammer-forged barrels since 1991. They may not be on par with high end match-winning barrels, but most have done well, and they're pretty consistent across-the-board.
Wilson barrels can be found on tang safety 77s with prefixes 79, 770, 771, 772, and 773. (773 also including the transition to hammer forged, but there weren't many.)
Douglas barrels can be found on tang safety 77s with no prefix (just the serial number, up to 4 digits), or prefixes 70 through 77 (skipping 76).
Tang safety 77s with prefix 78 were special runs, generally reserved for varmint, target, or special edition builds. Because some of these actions were used to build rifles from the early '80s into the mid-nineties, they can have any of the barrels installed on them. ...And Ruger almost never gives out ship dates for 78-prefix rifles, so trying to identify the barrel won't happen.
Mk IIs with the 780 prefix included transition time from Wilson to Ruger barrels. They should be avoided, unless you just want the action.
The rest of the Mk IIs have the best barrels Ruger has put on their rifles - with quality improving as time passed.