For me it changes a lot depending on what I am making but a good start up set ought in my view have the following:-
A long 1/2" straight cutter, preferably a replaceable blade one.
A bearing guided rebate cutter with a few sizes of bearings
A flush trimming bit, preferably two though, a top guided and a bottom guided
Chamfer bit
A 1/4" round over bit
A 1/2" round over bit
A 1/2" cove cutting bit
A bearing guided slot grooving set, adjustable to suit various groove sizes or to form tongues for tongue and groove boards
A smaller straight cutter 1/4" or 3/8"
Depending on what you plan to make the list will change and you will of course add to them over the years as projects demand.
Rob has a good list. I like spiral bits over straight bits for grooves and dados as I find I get cleaner cuts, especially holes for mortises. I would add an ogee bit to the list and maybe a core box bit instead of a cove bit so that you can make different types of molding -- most moldings are coves, ogees and round-overs. You might want to add a dovetail bit for sliding dovetails too and other dovetail joinery. I have added bits as projects dictate and I have well over a hundred bits in different sizes, profiles, etc... Limiting it to just 10 would be a hard choice.
Definitely have to have at least one spiral- up... All the better to bore with....
I also use a hybrid spiral up/ down for crisp melamine edges...
I do a bit with plywood so I always have a 1/8 round over in the trimmer...
Sadly I've never seen a set that I've looked at and gone "WOW" to the combination or the quality [sad]
Definitely rough plan a few different types of projects and consider the bit requirements to make should you isolate the right ones.
I'd personally treat specialist/jig-specific bits as outside your "magic 10" ... I think your 10 will quickly grow to 15~20 without considering jig specific bits.
(You tend to buy the special ones along with something else)
Find a few only catalogues of router bits and get familiar with the terminology.
If you are just starting it might be worthwhile to buy a set that has ten to twelve of the "basic" bits. You will find out that you will add to your collection as each project dictates. I tend to buy top quality bits if it is for a certain project because you will usually find another project later on that you can use it for.
I also have, and like, bits from Festool, Whiteside, and Freud (not Diablo.) Freud has the new Quadra-Cut bits that look interesting: http://www.freudtools.com/t-quadra-cut.aspx
I would recommend getting a good router bit set and then adding more as you need/want them. Everyone that has replied has given you very good advice as well.
I agree with John, I've never seen a set that had all the bits I use and they all seem to have some bits I'll never use. I really don't see the appeal unless you are getting a significant discount to buying them individually.