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I do not have a copy of the website on my personal computer
BAD! GET ONE!
What is the best way, or best software to use
FTP
Find yourself an FTP program(free or paid) and then download your site to your pc. Then, FTP into your new host and upload the site there.
Once you are ready for the switch, you go to your domain name registrar's control panel and change the DNS to point at the new server. It'll take up to three days for the switch to take place from there.
Hoprfully both hosting companies have provided you with the info you need to ftp
Regards
Is your site scripted, or is it just html? If it's just html, you can download a tool like wget to pull down your site.
Do you have shell or remote desktop access to the site your host is on now? You should be able to log in and backup the entire folder structure of your site from there as well.
PM me the url of your webhosts(old and new), and I will look and see if there's any other suggestions I can make.
I would recommend setting up your website first with your second provider and test if everything is running well. Especially if you are using things like database access etc.
Then changing the DNS settings causes all users to be redirected to the new location and your site will be accesible during this process.