The first, and most important, thing is to bury the pipe deep enough. For a driveway, that means at least two feet deep.
This, in turn, means you will need a hole on either side of the drive at least three foot deep, two foot wide, and four feet long (extending away from the edge of the drive. At the end away from the drive you should dig an additional bucket-sized "sump" for the water to drain into.
For the actual bore, I use 24" lengths of black pipe (3/4"), joined with electrical couplings (sure, the leak some, but are easy on/easy off). At the end of the first pipe is a simple brass nozzle, the type that looks like an old fashioned fire hose nozzle, and makes a sharp stream of water. The last section of pipe has a fitting that fits on the garden hose. Last- but certainly not least- I screw a ball valve on the end of the hose.
Ordinary water hose pressure is enough.
The procedure is simple....bore, stop, add another length of pipe, repeat as needed.
You will get wet and dirty. I usually work from one side untill too much water accumulates- then I move to the other side, and so on. This is also a good time to shovel out the mud that has flowed from your bore.
I say dig deep for two reasons. The first is that the drive actually shifts every time it is used, and this will eventually damage your pipe.
The other is that you will undermine the ground that supports the drive....which will lead to the drive cracking there.
So please do yourself a favor and resist the temptation to run the pipe just under the drive. The holes on either side of the drive- well, the bigger they are, the easier your job will be. If there was a hole specifically designed to be dug with a Bobcat, this would be it!
It will also probably be easier to run the pipe if it is cut into shorter pieces, and assembled as you push it in. Forget tape- glue a pipe plug on the leading end, and cut it off after you're through. Use as large diameter pipe as you can- even if this part is larger than the main pipe run- for your ease of pulling (as well as filling the bore you made).
Where your pipe exits the drive, use a sweep (elbow) to come up into an "in the ground" type of box. These boxes are usually round, grey, plastic, and about the size of a 5 gallon bucket. Your other pipe can enter the box from the side.