Auto-ripple is your friend. Turn that on and then when you make space or remove space everything farther down the timeline moves with it automatically.

You can select a bunch of clips by Ctrl-clicking on them, then move them all by moving one of them. This is good when you're trying to move a bunch that aren't all together.

You can also right-mouse-button click on a clip and choose 'select events to end' if you want to move everything from that clip to the end.

There's also an insert / time menu option that inserts space at the cursor and moves everything after it.