Lots of good knives out there. A 7" (18 cm) Santuko knife would be a decent choice if you want just one knife. Mercer is a decent brand ...
Yes. This is good advice.
Personally, I prefer a more traditional chef knife shape, say 6-9" blade, over a Santoku, but I use both. The Mercer knives aren't at all bad for the money. The budget Mercer 8" chef's knife can be had for about $15 and will come very sharp. Use a cutting board, hand wash it, and store it where it won't bang into other knives and you shouldn't have to resharpen too often in a home kitchen. I find most people take the edge right off a knife by cutting on plates, counters, etc. A ceramic plate is much harder than the steel. Just use a wooden cutting board, and store them in a block or with a plastic guard.
In the budget realm, Victorinox are also good, but a bit more than Mercer. A lot of professional chefs are using Mercer and Victorinox knives. Compared to most home cooks they have them sharpened more often.
There are some great Japanese knives, but they tend to be more expensive, have thinner edges, and be heat treated to a higher Rockwell hardness than most European knives. This makes them superior for cutting soft things like vegetables and most meat, but also more delicate. If you hit a bone with the edge or don't carefully store them they will tend to chip rather than just mash the edge.
I've been sharpening up a few knives for Thanksgiving carving. I suggest most people get an inexpensive Norton dual sided stone (Crystolon/India) or a budget Japanese synthetic waterstone and learn how to maintain the edge. A decently sharp knife will make even slicing through a soft heirloom tomato easy.