I've had a bike in indoor storage (my garage) for over 10 years. I'd like to get it back into good condition to use for my commute to/from work each day.
What things should I check/replace before I trust it to safely and reliably work on the road?
If the bike was in decent shape when put away, and has not been exposed to the weather, there's probably very little that needs to be done.
The tires would be the biggie -- likely they're checked and may need replacement (though not for certain). Inflate them to listed pressure and check the sidewalls for cracks or bulges.
Assuming rim brakes, the brake pads may very well be hard and in need of replacing. But you can't tell for sure until you try.
Otherwise, wipe off the dust, inspect for rust, and put a bit of oil on the chain. Check that the wheels spin freely, with no play in the bearings, and that the steering tube likewise moves freely with no play.
Give it a test ride. Make sure the brakes work OK and that the wheels don't rub as they turn. Try a few corners to make sure there are no odd handling issues.
The shifters likely will be balky at first -- run them through their paces a few times.
Basically, if the tires are sound and the steering is good and the brakes are good and there's no massive rust anywhere, there are likely no major safety issues. Dependability then becomes your concern, since you will be commuting, and there it's mainly the condition of the tires and how well the shifters work.
[I did miss mentioning cable housings. The cables should be given a shot of spray lube down the housing tubes. This is often all that's needed to turn a balky bike into a smoothly shifting/braking one.]
Your local bike store (LBS) can be an expensive solution in search of problems, if you follow my meaning.
Let me clarify: a few years ago I bought a mid-1980's vintage mountain bike which I intended to use for basically the same purpose the OP described. It cost me $20 at a garage sale. It had been a few years since I had been on a bike, so though I once knew how to change a tire, etc., I had long since forgotten. I don't know why, but it never really occurred to me to look on YouTube, ask on Bikeforums, and so forth. Long story short, my LBS charged me nearly $250 for a bunch of easy stuff I could have/should have done for myself: new tires, new tubes, new brake shoes, a little lubrication. In hindsight, it was my fault; basically I approached this like an idiot with an open wallet. It would have been worse, but I stopped them from doing a bunch of other stuff which was unnecessary and even more expensive (changing cables, new freewheel, blah blah blah). I still have that bike, and nearly 1500 miles later, I have a clearer understanding of what it needs and does not need. It didn't need all the expensive stuff they told me it needed.
If you're going to commute on this bike, you will need to remember/relearn a few things: changing tubes/tires, lubrication, cleaning, etc.
So while I agree the folks at the LBS can be a great resource, it's not always true. Before you go throwing money at this, do your homework a little. Post some pictures of the bike, either here or at Bikeforums, and include some closeups of drivetrain components: front and rear derailleurs, brakes, chain, rear cassette/freewheel, tires, and so forth. Include some specifications: are you the original owner? How many miles on the bike? etc.
Yes, some bikes probably require the extensive (and expensive) servicing mentioned by other responders. But I have a few bikes I've gotten used and I've put some miles on them, and in every case but one, the only things I did were new tires, new tubes, a little lubrication, new seat, new brake pads/shoes, and some cleaning. For the record, my vintage mountain bike (1986) is still running on the original grease in the hubs and bottom bracket, but it was evidently not ridden much by the prior owner(s).
You should probably take it to your local bike shop. An experienced eye will be more helpful that advice from the internet
By descending importance, my suggestions are
I think if you have to ask what your bike needs to become roadworthy, it is probably a good idea to take it to a shop of some kind. Doing things like re-lubing (or re-packing) all bearings, cables, replacing brake pads, adjusting stuff, etc... all adds up to a lot of work. If you haven't done it before it will take a long time, require tools, and you'll end up with less than optimal results.
What may not be obvious is that there are different kinds of bike shops. The ones that cater to enthusiasts (for lack of a better word) will tend to be more expensive and will push for pricey components and service charges. If you have a bike that is "worth it", you probably already know how to service it yourself or don't mind paying what it costs for a mechanic to fuss over it.
There are other bike shops, especially in big cities, that focus on commuters and used bikes. They have lower prices for everything and understand the point of view of somebody that wants to "bring back" a garage bike from a decade of idleness vs someone that wants to race in a weekly criterium.
As for what it "should" cost... something like US$ 50-80 for tune-up labor + 50-100 for tires and inner tubes (depending on quality) + replacement for things that are worn or rusted.
A good resource for parts for an old bike is loosescrews. If you want to do-it-yourself that is a good source, or you can just use it to make sure that you're not paying $100+ for a $10 dollar part.