I am not sure where to start here. First yes you
can use Linux on WiFi, but I think first you need to find out what
version of Linux you have on the new laptop, is it Ubuntu, if not it
isn't such a big deal, most Linux OS's are a lot alike. If it is Ubuntu
it would be nice to know which version. If it is very old you may want
to get a new live CD to bring it up to date.
Dial up may be an issue. Getting new software in the form of updates to
the OS will take a long time, they tend to be fairly large. Then the
question is can you get to a faster internet connection, like at the
library that has free WiFi connected to a cable connection.
Lets start there.