Depends on what you mean by 'often'. Also, 'how long'.
1) Have a land phone line connection and connect over a modem when phone service is available. (cheap, slow, not universally available)
2) Use a cell phone (More expensive, slower, generally available)
3) Use a wireless net (faster, limited availability)
4) Use a satellite system (very expensive, faster, universally available)
While there are ways to connect to the internet via cell service, the best is probably the plan from Verison, which seems to work well in most areas that are populated. But it is about the same as wired service. To price it, visit the Verison website, and while you are shopping you may want to also visit the site of Cingular to see what the have as well.
The use of a land line cost just depends upon where you happen to be. Most common for phones in RV parks is the use of a phone in the office which is usually free, but it is only available for brief periods since it is shared with everyone in the RV parks. As far as phone lines to your RV, there are parks that have "instant" phone lines to each site, but they are not very common and are mostly found near the larger cities. But more of the newer RV parks are installing them. Cost for the daily use when available seems to range from free, to as much as $.50 per phone call. And much more common in parks is the availability of a phone line which you pay the fee to connect to the local phone company and then pay the monthly bill. Of course, this service is not usually practical unless you plan to stay in that park for two or more months.
The best, but most expensive choice is the use of a "dish modum." There are two versions of this. There are the automatic dishes that are installed to the roof of the RV and which will locate the satalite and get you on-line with little help from you. These units cost about $4500 installed and then a monthly feel of $60 - $100 per month. The second way to get a dish connection is to go with one of several dealers who now have FCC approval to seel a portable dish with transmitt/receive capabilities. Those cost about $1500 for the equipment and the installer has to train you to set it up and to point the dish which is slightly more difficult than the setting up of the standard TV satalire dish. The monthly cost for that service is about $60 per month.
Pretty much answered above, but to reiterate:
1) Depends on the RV park, whether it is available and how much. I suspect one of those big books listing 'all' the campgrounds in an area may say, so choose the ones you like and then call ahead to insure they have it available, and the cost.
2)If you get Verizon, and you compute at night and over the weekends, you might be able to get away for as little as $40 a month. It should work in about 95% of the country, best in the Eastern half of the US, almost as good in the Southwest, and mediocre in the Northwest. Depending on your phone, you may have to buy an interface card for your PC, perhaps $50 to $200. Don't know about any other provider
3) Don't have a clue. You would need a WIFI enabled computer and then check with the RV parks you were thinking about staying at. I'm thinking it might be 'free' or reasonably priced, but perhaps fairly rare at this point in time.
4) Last I heard, it was $100 a month after $2500 or more installation. Of course, bound to come down over time.