For the barrel, running a bore snake down it will keep it clean enough. I also run a cleaning patch with Break Free down the barrel a few times to keep rust away. A word of caution on break free too, just a few drops go a long way.
Another good idea is to get a toothbrush (I just use one that I wore out instead of throwing it away) and put some gun cleaner on it and scrub out the chamber. Then go over it with a patch of break free for rust protection. If your gun has interchangeable choke, then you should also unscrew that and wipe all the grime off it inside and out, then of course, coat it with something like break free. I will usually do these 2 things after every few times out shooting.
And if it seems like I'm using way too much break free, it is becasue I live up in Western WA which is a very wet and humid place. I've had guns start to rust in my trunk if left in there for more than an hour. My friend has also had his choke tube rust to his gun before and it took him a lot of work to remove it without damage to the threads. So for us an ounce of prevention is better than a pound of care. Plus it isn't that break free is the only thing you can use either, it's just been my favorite after trying a few different kinds of the like. There was actually a nice test done at
http://communities.prodigy.net/sportsrec/gz-rust.html IMO it is worth reading.
For the outside of the gun, wipe it down a large cleaning patch with a few drops of break free and then rub off the excess with a dry cleaning patch.
This will get all the dirt and body sweat off. I do this every time I clean my gun.
I've never even seen, let alone owned a version of the 37, so I can't tell you whether or not you need to disassemble the action. My browning BPS has never needed it to my knowledge plus the owners manual says not to take it apart anyway so I haven't. Some here recommend taking the gun to a gunsmith and have him clean it and completely go over it, but I've never tried this so I can't comment on it's worth.
If you have a wood stock and really want to go the extra mile then you could always buy some paste wax and wipe it on and buff it off of the stock. Plus a $5 can of this stuff will last you darn near forever. I'll only do this once every couple of months with my wood stock guns.
Sincerely,
Adam