I have one old computer that was given to me that could possibly be connected to the internet, and my personal school computer. I'm using a listening center and love that, and an overhead projector, but I don't know what to use on the computer.
At the kindergarten level the applications that I use include Kid Pix and http://www.starfall.com Starfall is fun to project on the large screen if one is available to you in your building. Students at this level enjoy google earth as well. You can put Google Earth on the large screen and 'fly to' places like the Eiffel Tower, Disneyworld, to visit the 7 continents, travel across the oceans, over the mountains and across bridges. It's great fun as a whole group activity. I take photos of the students and put them into http://www.voicethread.com then have them record what they are doing in the photo. Another website that I use with the students is 'Roy the Zebra.' http://www.roythezebra.com At the kindergarten level I read the story about Roy the Singing Zebra aloud to the students. In first grade the students continue with Roy the Zebra interactive games.
I think the simplest way just to get started is to put a cd rom in your computer of a kid's game. Lots geared to Kindergarten You can find these really inexpensively or at the library. Sounds like internet is iffy...and cd rom games are a good way to start them using the computer, especially if you are new at this too. If you are able to connect to internet, I agree that starfall.com is a great site.
A couple of things come to mind. One you can use the keyboard to teach students the alphabet and letter/number recognition. One way to do this is to print out a copy of a keyboard with letters and symbols or without (even better) and as the kids learn them have them place them on the keyboard. We did this at one of the elementary schools I worked at and called it the kids "laptops". This helps later on with keyboarding, etc. Another idea is to have students work with your overhead for show and tell. Have them use it to show off what they are sharing so they are familiar with using a presentation device. You can also have a class play based on a book you are reading to them, have them act out scenes from the book to be taped and posted to a website. You can also maintain a kindergarten blog where the students are bloggin their learning and you are acting as the scribe.
I agree with Lorraine - Google Earth can be used as a starting point for a lot of your class activiites. For instance, when celebrating the Chinese New Year, you can use Google Earth to show them where China is compared their school. You can quickly find the distance, and even zoom in to a particular city from a story. In just a few minutes you can give them an impression of their place in the world, more than the standard classroom globe can. You'll get lots of good ideas on Classroom 2.0 and elsewhere once you start looking.