JP,
Keep the soil slighly moist, but not wet. Make sure the seeds stay in a warm place. Ideally, use a seed starting heating mat to provide heat from the bottom. Otherwise, just put them in a sunny spot or on a refrigerator - somehwere that gets warm.
It may take a few weeks for them to sprout. When they get large enough to handle without being too delicate, you can transplant them to individual containers. Use a very free draining potting mix - I use about 40%cactus potting mix and 60%perlite to start - it drains very well.
Citrus like lots of sunshine, but try not to scorch the tender young ones.
Also just be aware that it will be at least 2-3 years before your limes will flower and fruit - possibly longer, and the lemons may take even longer as much as 5 years or more.
If you want a tree that will bear fruit soon, think about buying a grafted tree. Meyer lemons and key limes in containers are available this time of year at garden centers in most parts of the country so you shouldn't have much trouble locating one.
Good luck!
Pelham
Good luck