Ok, so I've heard that you should tile after installing cabinets.
Regardless of what you heard, you can either tile first and then install the cabinets over the tile, or install the cabinets and then tile up to them. The only thing you SHOULD do is caulk between the cabinets and the tile if the cabinets are set on top of the tile. The reason why is that tile is set by hand, and so the tiling won't be perfectly flat, so it'll be necessary sometimes to shim between the cabinets and the floor. (Often it's necessary to do that anyway, regardless of what flooring you have.) You need to caulk any gap betwixt the cabinet and the floor because if you spill a bowl of soup near the cabinets, you'll create a bug's idea of heaven. The soup can run under the cabinet and bugs instinctively protect themselves by crawling into as tight a crevice as they can manage. The reason why doing this makes sense to a bug is that if he's in the smallest crevice he can fit into, then no bigger bug can get in there and eat him. So, a cabinet that's just a millimeter or two off the floor with plenty of food under there is sure to be discovered by bugs, and it'll make the ideal breeding ground for them to have lots of little buggies that all look just like dad. Caulking that joint will keep the food out, and then there'll be no reason for the bugs to want to live there.
I'd caulk that joint regardless of whether the tile or cabinets came first.
If we tile after the cabinets are installed then would that mean we can't tile under our appliances?
No, you set the tile in those places before sliding the appliances into place.
It seems like if we did tile under our appliances that they would then be too high (for instance would the dishwasher still fit under the countertop) and if we don't tile under them, then wouldn't they be stuck in place making it hard to remove them for any kind of repair or replacement?
It only SEEMS that way until you discover that all built-in dishwashers will have adjustable front and rear legs so that you can adjust the height of the dish washer to accomodate flooring of different heights. For example, linoleum versus ceramic tile. Typically, the front feet on a built-in dish washer will screw up and down, and there will be threaded rods on each side by which you can adjust the back feet. Typically, you NEED at least 4 to 6 inches of space above any fridge to allow the heat from the condensor coils (the black things on the back of the fridge) to convect into the cooler air in the room. Fridges will typically draw cool air in from a grille at the bottom of the front of the fridge, have that air rise as it flows over the condensor coils at the back of the fridge, and then allow that warmed air to escape through a gap between the top of the fridge and the underside of any cabinets above it, thus creating a convective current. And, that gap is big enough to allow for even the largest differences in floor height.
What you SHOULD do if you have a prefabricated plastic laminate counter top is paint the underside of the counter top over the dish washer with oil based paint. The reason why is because when you open the dish washer and all the steam comes out, you don't want that steam condensing on the bare particle board. If it does, it will be absorbed into the particle board and cause the particle board to swell. Often you see the laminate on countertops loose right in front of the dish washer and also often in front of the kitchen sink. The reason for this is that the particle board has absorbed moisture and swollen, and that swelling of the wood has resulted in the glue failing between the laminate and the particle board. You totally eliminate this problem by preventing the wood from absorbing water and swelling, and you can do that with a coat of high gloss alkyd paint applied to any wet areas or areas that will be exposed to steam.
I would like to save the money and not waste all the extra tile going under the cabinets, but I'm trying to figure out how we get everything at the right height then.
Didn't you learn your lesson with the Pergo flooring. ALWAYS ALWAYS ALWAYS buy extra flooring in case you ever have to replace some. If you were to drop a stainless steel pot on your kitchen floor, you might crack a tile and might want it replaced. If you were to come home pie-eyed from a night out with the boys and tried to juggle bowling balls just like Frank does, then you'd be likely to need spare tiles. Or, if in the future you decide to take out the wall between the kitchen and dining room for a more open look, then you're gonna need spare tile. Spare tiles are cheap when you're buying flooring, but they're priceless once you can't get them anymore.
Also, reconsider "Ceramic" floor tiles. The difference between ceramic tiles and porcelain tiles is that porcelain tiles are the same material all the way through the tile. So, if you drop a knife on a ceramic tile, you might chip the tile, in which case you'll see the difference colour of the "biscuit" of the tile (which is the fired clay body of the tile). If you do the same with porcelain tiles, you won't notice it as much because the tiles are the same colour all the way through. However, porcelain tiles are more expensive and the ones I've seen are smooth as a mirror, and would be very slippery in a kitchen where the floor could be wet or have spilled food on it.
Also, both ceramic tiles and porcelain tiles come in "hardness ratings" of one to five diamonds. (I've never seen a ceramic floor tile rated at less than 3 diamonds, so maybe the lower ratings are for other kinds of ceramic flooring, like clay tiles. The harder the tile, the more diamonds it's rated at.
About buying tiles:
Both ceramic wall and floor tiles will be made in "batches" at the factory, and each such batch will be given a "dye lot number". Boxes of tiles with the same dye lot number mean the tiles all came from the same batch, and should be absolutely identical in colour. If you order your tiles from a ceramic tiling wholesaler or retailer, they will know this and ensure that all the boxes of tiles you get will be from the same dye lot. However, if you buy your tiles from a hardware store or home center, then you'll want to inspect each box you buy to ensure it has the same dye lot number as all the other boxes. It doesn't matter what the dye lot number is, just as long as it's the same on each box.
(If you find that you can't find enough boxes with the same dye lot number in a home center, you're only two choices are to order the same tiles from a tile retailer, or buy tiles from different dye lots and mix them up before setting them so that the tiled floor looks like it was intentionally laid with tiles of slightly varying colour. Pay attention to the dye lot numbers because nothing is gonna pi$$ you off more than half your kitchen floor being one colour and the other half being a very similar colour.
Often there won't be any perceptible difference in colour between tiles from different dye lots. However, the only way to be sure is to ensure all boxes of tiles you accept have the same dye lot number. And, the dye lot number will be large on the box for quick and easy identification of boxes that are piled up on metal racks 15 feet off the ground. It won't be hard to find or read the dye lot number.
This stuff about dye lot numbers also holds true for other kinds of tiles as well, including vinyl composition floor tiles.
And, of course, you want to seal your tile grout with an ACRYLIC film forming sealer, not a penetrating sealer. Penetrating sealers are great for showers, but in a kitchen the greater risk is having a soft food like peanut butter or mayonaise mooshed into the grout where you can't get it out and will provide a food supply for bacteria. On a floor or counter top, you'
re better off with a film forming sealer that won't allow anything to penetrate into the porous grout in the first place. Also, there are two basic kinds of grout sealers; silicone based and acrylic. Silicone based grout sealers are more common and less expensive, but the problem is that you can't apply more silicone based grout sealer to silicon based grout sealer; it just won't stick well. And that's true of all silicates; you can't get new silicone caulk to stick to old silicone caulk. Acrylics stick well to each other, just like one latex paint will stick well to another. If you're looking for a good acrylic film forming sealer, here's the one I get from California, (import, actually) and am very happy with:
http://www.glaze-n-seal.com/sealers.html
http://www.glaze-n-seal.com/docs/factsheets/Grout Sealer FS41.pdf
Hope this helps.