Thanks for getting back. Outlook 2013 should save all your emails that you currently have. Before you remove Office 2007 copy your Outlook .pst Files into another folder, so in case something goes wrong you will still have a copy of these. Office 2013 stores these files in your Documents Library under Outlook Files, but I am not sure where Outlook 2007 stores them so you might need to hunt around for them.
This website may help you locate your data files & help you with the setting up of Outlook 2013.
How to move Outlook Data to a new PC | PCWorld
Just to be sure that you have all the info at hand in case logging on doesn't work when you finally get installed, write down your 2007 Account Settings you can access these through Control Panel & click on the applet Mail Outlook 2007 & when that opens, click on Email Accounts, then click on Change to get access to your existing settings & follow through as needed.
Write down all your Outlook 2007 Account details, which covers items like your log on details with your email provider, things such as Account Type, for example is it a POP 3 account, Incoming mail Server, Outgoing Mail Server, then click on More Settings & write down all the Settings that you currently have. You will need to set all these in Outlook 2013 & it should work. If you have difficulty with getting Outlook 2013 to work it is worth phoning your provider & asking for help.
It does sound very complicated but actually it should not cause you many problems.
If you wish to remove Office 2007 before you install Office 2013. Read through this MS information as Office can sometimes be hard to get completely uninstalled.
https://support.microsoft.com/en-au/kb/928218
Feel free to get back if you have any questions or bump into any problems.