Here's a super easy example:
The track is Wild Horses by The Sundays. I currently have it on "Buffy the Vampire Slayer: The Album", a compilation from 1999.
The easiest ways to get tracks into Picard from MB are either drag & drop or to configure Picard as an external tool (see first post in this thread).
Unless you've tagged it with Picard previously, the file should appear in the left hand pane:

You should see two buttons in the Picard toolbar labeled "Scan" and "Lookup". Scan looks things up based on the audio fingerprint of the file, Lookup looks them up based on tags. For this purpose, Lookup will usually serve you better.
I clicked Lookup, and based on my preference for albums over compilations, Picard correctly loads the album Blind by The Sundays, from 1992.

I did this one with the original album info protected. This is the list of tags I put in "preserve these tags":
album, albumsort, title, titlesort, artist, artistsort, albumartist, albumartistsort, tracknumber, totaltracks, discnumber, totaldiscs, date, label, compilation
This is what I have in scripts (if you have info in them, you could protect them instead):
$unset(media)
$unset(barcode)
$unset(catalognumber)
$unset(script)
$unset(asin)
In the bottom panel, you should see the old info and the new info for the file. As you can see, most of the info hasn't changed:

Looking further down, though, you can see that it has picked up the original date/year from the album, as well as some other info:

Okay, so maybe we need to protect or unset Release Country, Release Type, and Release Status, too. But you get the idea. And picking up the personnel from the album might be desirable! (You will also see a lot of "MusicBrainz" tags which you will have to decide how to handle. I suggest keeping them in case you need to look something up again, but if your compilation tagging was also with MusicBrainz you might want to keep the original value instead.)
---
This will get a little tougher if Picard doesn't find the right source album right away, or if you're not sure if it's the right source. If Picard had not found it (it is occasionally WAY off the mark, sometimes because the data was entered incorrectly, or because some combination of settings goes haywire), I would use the "Lookup in Browser" command to try to track down the source myself. This works best if the track is in the left panel. If it was associated with an incorrect album on the right, you can drag it back to the left.
Doing a browser lookup on this track gets me here:
http://www.musicbrainz.org/taglookup?tport=8000&artist=The%20Sundays&track=Wild%20Horses&tracknum=14&duration=286119&release=Buffy%20the%20Vampire%20Slayer:%20The%20Album&filename=12%20Wild%20Horses.mp3If you look in the right hand column, you will see that the track was used on some compilations, but there is also the original album and a couple of singles. If you know which one you want, usually clicking that green "tagger" icon will load the release in Picard. If it doesn't, you can also paste the URL for the release in Picard's search bar.
You'll also find the search parameters at the bottom of the page, and you can change them (for instance, removing the album title from the search).
---
Of course, if you get one of those tracks that is on a million compilations, it's going to be more difficult to track down. That's usually when you have to start doing some real research. Here's a more challenging one:
I have a recording of Caravan by Duke Ellington from a compilation called Big Band Fever. Doing "Lookup" just gets me another compilation, from 1993. Doing "Lookup in Browser" doesn't have great results, either, until I remove the album title from the search. Then I see a single (I usually look for singles of anything pre-1950ish):
http://www.musicbrainz.org/taglookup?tag-lookup.artist=Duke+Ellington+%26+His+Orchestra&tag-lookup.release=&tag-lookup.tracknum=1&tag-lookup.track=Caravan&tag-lookup.duration=169874&tag-lookup.filename=1-01+-+Duke+Ellington+%26+His+Orchestra+-+Caravan.mp3It's from 1937, but it doesn't have a length for the recording that I can compare to mine. Now I have to decide if I care whether it's the same recording, or whether I want to do the research to identify different recordings. You'll have to answer those questions for yourself.

---
There is always the possibility that the original source of the track isn't in the database. For most mainstream music since ~1970, that doesn't seem to be much of a problem. The earlier the release, the more likely you're not going to find it. I hope you'll think about adding it to the database, if you end up looking something up that's not in MusicBrainz. There are some really handy import scripts for discogs, bandcamp, CD Baby and so on.
---
Questions/suggestions? Specific cases I can help with?