I'd be surprised if the wires coming in from the bottom of the box (the 14/2 NM) weren't the supply (hot). It's fairly common to use a 14/3 for the run between the switches and the light\fan. In that you only have 2 cables coming into the box and you're working with single-pole switches, the easiest way to verify is to use the regular switch to test for the hot.
I'd connect one of the non-green terminals to the red wire coming in from the top and one of the non-green terminals to the black wire coming in from the bottom. Switch it on, and turn the breaker back on. My guess is that either the fan (most likely) or light will come on.
You can then confirm this by turning the breaker back off, removing the red wire from the switch, and replace it with the black wire coming in from the top of the box. Turn everything back on, and the other device should be on now.
After you've confirmed that the bottom black wire is the hot, follow the installation instructions on the new switch and dimmer. You'll need to connect both of line terminals on the switches to the hot. You can either jumper them (tacky) or wire-nut two pigtails onto the hot and connect one to each switch.
The simplest thing to do is to turn the breaker on and figure out which black wire is supplying the current. (This is arguably one of the more dangerous approaches, but it is safe if you follow precautions like have someone guarding the wires so they are not touched while the power is on.) Use a multimeter and test the voltage between the ground wire and every other wire. Exactly one black wire should have 120 volts on it.
If not, stop and report back here.
Once you identify which black wire is hot, turn the breaker off.
Then wire nut all of the whites together. Wire the "hot" black wire you identified to one other black wire. Turn the breaker on. Note if the ceiling light came on. If not, turn the breaker off, hook up the other black wire to the hot black wire and turn the breaker on. Note if the ceiling light came on. Turn off the breaker.
Now you know: 1. which black wire is the "hot" wire, and 2. which black wire goes to the ceiling fixture, and 3. your "other" black wire, which probably provides power to another switch or outlet.
You should wirenut the "hot" wire and the "other" wire together with a short scrap of black wire to create a pigtail. Connect the pigtail to the switch. Connect the black fixture wire to the other side of the switch. And your whites should already be wire nutted together from before. So just hook up the ground wire to the switch, and you're done!
Best Answer
Yes, but if there is a ground wire in the box, you should attach it to the ground screw on the switch.
If there is no ground wire, but the box is metal, you should connect a ground wire from the switch to a screw put into one of the threaded holes in the back of the box, or use a ground clip placed on the edge of the box.
If there is no ground wire and the box is plastic, you can ignore it.