Bicycle Tutor

I have a set of new 3/16" bearings that are just barely smaller than the bearings I need to replace. Is it ok to use them? Do I have to use the exact same size?

Probably, but I wouldn't unless you have to. You can often get away with using a slightly smaller bearing in a hub. Especially on lower end parts where the tolerances aren't that precise anyway. But you are putting in parts that the hub was not designed for and changing the way everything lines up against each other. I'd spend the couple bucks to get the right size.
Beater bike - do it. Bike you want to last - get the right ones.

Hi! I have just overhauled my front and rear hub on my mountain bike. I noticed though that the cones had some pitting. I asked about getting some new cones at my LBS but they don't carry them. They just said that I should get new hubs. What can I expect to happen since I just decided to put in new bearings and grease? Is it just added friction? or are they more likely to crack in the pitted areas?
For reference, the bike is a 1994 specialized rockhopper that got tons of use before I left the country 7 years ago. Since returning, I have used it for commuting and trail riding and want to start touring.

It is added friction, in my opinion: lots of it. It will also make setting up the hub more difficult. Replacement cones are almost impossible to find and since entry level front hubs are around 10€, it is economically not feasible to replace cones.

The pits in the cones will tear up the new bearings reasonably fast. And add friction, difficulty of adjusting, etc. Should be fine for the short term, but you definitely don't want to start touring until you've resolved this.
It is true that hubs are cheap, but you still have to rebuild the wheel. What I have done in the past is to buy a complete similar hub and take the axle parts from it to rebuild the one on the bike. Seems like a waste, but it is true that it is hard to find replacement cones, and you can often find a hub pretty cheap.

Thanks for the posts guys! The adjustment was difficult. I kept going back and forth between a tiny wiggle in the wheel to just tight enough that the hub no longer would oscillate by the weight of the valve stem alone. I am considering getting those new hubs and building the wheels my self. I was just on another site about building a wheel truing stand. The problem will be proper dishing. Anyone have experience with hubs for touring?

I can definitely recommend http://www.wheelpro.co.uk/wheelbuilding/book.php In the book are also plans for building a truing stand and some neat little tools (nipple driver, dishing gauge). I own a Minoura truing stand, it is ok (and was inexpensive). Getting the correct dishing is not difficult, if you have a dishing gauge. Mine is self made (scrap wood). It is shaped like this: _/^^^^\_ The "feet" (_) are level, about 40cm apart, 15cm long and 7cm tall (base of upper part (marked ^^) to bottom of feet). I first just held a spoke to the (^^) but now made a sort of "feeler" with some wood. Usage: you put the "feet" on one side of the rim and move the feeler so that it just touches the lock nut. Then (without changing teh position of the feeler), you put the gauge on the other side of the rim. If the feeler just touches the locknut, the dish is correct.
I find getting the wheel radially true to be the most challenging part of the whole process.
For touring I'd probably get mid-range Shimano mountain bike hubs (LX, SLX) or other mid-range mountain bike hubs.