Tunneling under a sidewalk isn't difficult, but the best way to do so is far from obvious. Most imagine they have to dig their way through or even jab a tunnel through using a straight pick. These approaches might eventually work, but they involve far more work, far more digging, and (probably) far more profanity than is necessary. A better way is to hook a piece of PVC pipe up to your garden hose, and blast your way through using water pressure.
What You Need
Assuming you already have a hose, you also need one piece of 1-inch PVC pipe about eight inches longer than the width of your sidewalk, and another that is a couple of inches longer than the sidewalk's width. If your hardware store offers more than one grade of PVC pipe, choose the one with the highest pressure rating. You also need an adapter to connect the PVC pipe to your hose and PVC solvent cement. Make sure the adapter is specifically designed to connect to a hose because threaded PVC couplings won't mesh with the threads on garden hose spigots. And if you don't already have a hacksaw or reciprocating saw, pick up a cheap hacksaw. PVC pipe cutters won't work, because you need to cut one of the pipes at an angle.
Putting it Together
The trickiest part of assembly is making the cut at one end of the longer PVC pipe, which is the one doing the digging. You want to cut at a 45-degree angle so that the PVC ends in a sharp point. The rest is just a matter of cementing the PVC to the adapter and attaching it to your hose.
Trench, Then Tunnel
The first step in tunneling under a sidewalk is to dig a trench perpendicular to the sidewalk. You want it to be deep enough to let you tunnel several inches below the bottom of the sidewalk and as long as the PVC pipe you attached to the hose. Dig another hole about a foot wide and slightly deeper than the trench on the other side of the sidewalk, where the tunnel will emerge. Now turn on the water and use the point of the PVC pipe to begin tunneling. By alternating in-and-out and twisting motions, you should make rapid progress. Pull the pipe all the way out from time to time to allow dirt to wash out of the hole. After breaking through, pull the PVC pipe all the way out, then push it back through several times to create a little more space. Finally, push the shorter piece of PVC through the hole and leave it in place to prevent the tunnel from collapsing.
For Bigger Tunnels...
If you need a tunnel that is wider than 1 inch, follow the steps above. You should still make your "tunneling" pipe out of 1-inch PVC, but the shorter length of PVC should be the width of your desired tunnel (but at least 1 1/2 inches). After breaking through and widening the tunnel as described, slide the pipe attached to the hose through the larger pipe, then repeat the tunneling process pushing both pipes through the hole. It should go much faster this time. When you break through, simply leave the larger pipe in place. For much larger tunnels, you can widen in stages by sliding progressively wider lengths of PVC over each previous one.