Installing panel off the meter box through 60A breaker
I think you're 90% "on it", but there are a few yellow-flags I see. Don't infer that I think it's a bad plan; like I say you're 90% there. 

"60A panel" - Go BIG.

The future happens. A very common question here is "I ran out of breaker spaces in my panel. What can I do?"  And that has a sad and expensive answer.  (Other than "Go back in time to when you bought the panel, and spend a few latté's more on a panel with a lot more spaces", but we're not allowed to say that.)

Since the panel is main-lug (no main breaker), that means "60A" refers to the busing.  That implies this is a very, very small panel.  So let me tell you now: Spend a few latté's more on a lot more spaces.  

Beware "X space / Y circuit". The "circuits" number is a lie. It relies on "double-stuff" breakers. NEC 2014, 2017 and 2020 have wiped those out, by requiring AFCI and GFCI on so many circuits. Those aren't available in double-stuffs.  

"Gotcha 5x5. I'll use this 18-space panel. But it's rated 200A! My breaker is 60A."  Right, the 60A is an absolute amp limit on the panel. Like HR-rated tires are rated 112 mph, or CRJ-200 airliners are rated for 41,000 foot altitude.  That doesn't make it a best-practice! 

So you are absolutely free to feed a 100, 125, 200 or 225A panel from a 60A breaker.

60A on #6 copper THHN aka THWN

Actually, because you are running THWN, you are allowed a 70A breaker.  (Technically you are allowed 65A, but you are also allowed to round up to the next offered breaker size). 

Also, #4 aluminum is the right stuff; on a heavy or long run, copper is a waste of money, with two exceptions: a)  if you have to splice in a box, rules for #6 are easier to follow than #4. And b) it may require larger diameter conduit, but PVC conduit is so cheap that aluminum is sure to win that one.  

There's some ooga-booga about aluminum out there, but that was about a) the obsolete AA-1350 alloy, b) used on small 15-20A branch circuits with c) terminals not properly rated for aluminum.  In your case you'll be using AA-8000 on a heavy feeder, on aluminum-rated lugs, that probably are aluminum.  

Your ground wire only needs to be bare #10 copper or insulated #8 aluminum. Yes, I know they make prefab cables with ground -2 sizes from the conductors; don't use that as a guide because there's no Code basis for doing that. Shrug... maybe it's a UL requirement because it's a cable.  (a 1/1/1/6 might be too brittle?) 

Main breaker! Main breaker!

I almost forgot. You're specifying a main-lug panel which means no main breaker.  Stop and see if it's an outbuilding.  (if it's connected with a breezeway, it is not an outbuilding). Outbuildings require main disconnects.  

You can have a separate disconnect switch if you really want to, but usually, the most cost-expedient way to get a disconnect switch is to get one "built into the panel with a number on it".  I.E. get a main-breaker panel instead, and let the main breaker be the disconnect switch and ignore the number on it.  

Of course you want main breaker larger or equal than the supply breaker (otherwise you're wasting feeder).  Now some people get the idea "if I make them equal, the nearby one will trip first and save me walking". No, it won't, both because of Murphy's Law and the way trip curves differ somewhat on main vs feeder breakers.