If the grout is discolored, there are a couple of ways to clean or improve the look of the grout. There are several household chemicals that make effective grout cleaners, such as hydrogen peroxide and plain vinegar, and there also are commercial cleaners. If the grout is too dirty, you can cover it up with a grout colorant. A grout colorant essentially is a kind of paint designed for grout. Applying it is a tedious job, but can make a big difference in the appearance of your table. Even if the grout is not discolored, coloring it can make the table better match other furniture or match the color of the room.
If the grout in the mosaic is chipped or cracked, this is another flaw you can repair yourself. Mix up a little bit of a grout that matches the current grout color. There are several kinds of grout. If the crack is very fine, 1/8 inch or less, use unsanded, acrylic latex or epoxy grout. A sanded grout can be used if the crack is wider. Add a little extra water to the grout to make it flow better, then smooth it into the crack with your finger. You may need to clean out or widen a crack for best results; you can do this with a grout saw, or use a rotary tool.
If you have a broken tile, the best tactic is to replace the tile with an identical tile. Remove tile by breaking up the tile and grout around it with a hammer and chisel, then add a new tile with some thinset. This can be rough on the table, so be sure to provide support or, if possible, remove the part with the mosaic from the rest of the table, and put it on a sturdy surface. Mosaics often are unique, so you may want to repair the tile instead. You can repair many mosaic tiles with patching resin, an epoxy. It can be colored to match the broken tile color. Clean out the tile with a stiff brush, mix the epoxy according to the manufacturer's instructions, tint if necessary, then apply it to the tile.
Mosaic tiles normally are durable and waterproof, but if you plan on using the table, sealing the mosaic provides extra protection. You should, at minimum, apply a grout sealer to the grout, or a clear sealer over the entire mosaic. You also can apply a layer of a clear resin, or a plain, clear piece of glass to the top; the second option needs smoothed edges, or to have the edges sealed somehow for safety. These second two options can add weight to the table, so you may need to add extra support. A table with a mosaic top should have bracing between the legs for support, at the minimum.