So, you want to go to Egypt? While you can take an organized tour from Israel, it is far cheaper to travel on your own and you get the benefit of doing what you want on your own schedule. The easiest (and cheapest and way to get from Israel to Egypt is using the Taba border crossing. It's also the only way to get there by land. There was a border crossing near Gaza in the past, but due to tensions it's closed. Besides, you really don't want to go near Gaza. Trust me.
The border crossing at Taba is a fairly low-key affair, as Middle East border crossings go. You'll want to take a taxi from Eilat to the border, or you can take Public Bus #16. A taxi will run you in the neighborhood of 30 shekels from the airport or bus station. It is not practical to walk to the border from town. The best time to cross is early in the day, unless you are trying to meet an Egyptian bus. Be aware that the border does close in times of security issues and some holidays. Check ahead if either of these situations apply.
Your first hurdle will be exiting Israel. This is a simple matter and you shouldn't have any problems unless you have overstayed your visa or have lost your Israeli exit card. Make sure to have that card with you and out when you prepare to exit Israel. Next stop: Egypt! The Egyptian side is more chaotic and less efficient.
There are two types of Egyptian tourist visas: Sinai only and full tourist visas. The only type of visa that can be granted at the Taba border is Sinai only. It is valid for 2 weeks and is *not* valid for Ras Mohmmaed. If you are going on to mainland Egypt (or Ras Mohammed), you will want to get a full visa from the Egyptian Consulate in Eilat or at your Embassy at home. Assuming you are getting a Sinai only visa, you will fill out the paperwork and pay the fee, which ran about $20 when I last was there.
The paperwork and security process can take anywhere from 10 minutes to an hour, depending on the number of tour groups crossing and how long it takes to scan your bags. Be prepared to answer questions about where you are heading in Egypt, how long you will stay and how much money you have. If you are in need of Egyptian pounds, there is a Bank ATM inside the border post. The fee is fairly low and I recommend using it as ATMs in Egypt are hit or miss.
Once through the border crossing, there are a few things to remember. The first is that there will be a large number of taxi touts waiting to take you wherever you want to go. If you have crossed in time to meet a bus, they will invariably tell you the bus is cancelled or has already left or any number of stories. Don't believe them!
If you do need their services, haggle. It's expected and to do otherwise will result in you being greatly over-charged. If you would prefer to drive yourself, there is a car rental agency in the Taba Hilton. It's past the border post on the left. We used this service (the owner's name is Max) and had a great time driving around the Sinai in a Toyota Camry. Our fee was roughly $35 a day. The last thing to remember is to get your visa registered by the tourist police right away!!. Hotels will do this for you, but campers and those in youth hostels may want to do it themselves. There are numerous horror stories of people being charged exorbitant fines for not having their visas registered. Don't risk it!