Installing Wall (Upper) Cabinets
Starting in the corner, set the first unit in place and snug it up to the wall. Use a level to make sure that it's plumb up and down both sides and front. Shim as needed to hold it plumb.
To secure the first cabinet, pre-drill and drive a couple screws (2-1/2" drywall screws are usually fine) through the top and bottom nailing strips in the back of the cabinet.
Set the second cabinet in place, snug against the wall and snug against the side of the first cabinet. Line up the adjoining face-frames of the two cabinets and clamp them together tightly.
Then pre-drill through one frame into the other and secure them with a couple of screws. Shim the second unit as needed and secure that one through the back. Continue that way till all the wall cabinets are in place.
NOTE: If you're installing frameless cabinets, you can clamp the sides together as described above but you can't simply screw them together. The manufacturers will include special hardware to bolt the sides together as well as caps to cover the hardware.
Setting Base (Lower) Cabinets
Starting in the corner, set the first base cabinet and line up its top with the reference line. Shim under the back of the cabinet if necessary to raise it up to the line. Then screw through the nailing strip at the top of the cabinet into the wall studs.
Check the first cabinet to make sure it's level along the front and the sides, and shim underneath as needed.
Set the second cabinet in place and snug it up to the wall and to the first cabinet. Follow the same steps to secure the back to the wall studs.
Line up the back, clamp and secure the adjoining face frames as described above for the wall cabinets. Shim under the second cabinet as needed to line it up with the first unit. Continue in the same manner till all the base units are in.
Installing Filler Pieces
Once you've completed a line of wall or base cabinets, you can deal with any gaps left in the layout by fitting filler pieces.
Rip-cut the filler piece to the width of the gap with a table saw or a circular saw having a rip fence
Set the piece in the gap, filing or planing as needed to fit
Line it up exactly with the adjoining face frames
Pre-drill and screw through the adjoining frames to secure it.
Installing Island Cabinets
Installing island cabinets is a bit different since they're standing in the middle of the kitchen with no wall to back them up.
So the usual technique is to secure 2x4 blocks to the floor and screw into those through the cabinets' toe kicks to secure them.
The hard part is getting those blocks in the right spots. Set the units in place, line them up just where you want them, mark the floor along the outside edge of the toe kicks, then position the blocks inside those marks but offset the thickness of the toe kicks (usually 3/8" or 1/2").
Screw the blocks onto the floor, set the cabinets in place over them, shim the cabinets level and plumb, screw through the toekicks into the blocks and secure the frames together wherever two cabinets meet.