With the verb ENTER, I would use INTO: "Please enter all these names and addresses into my Contacts." With a verb like DOWNLOAD, INSTALL or LOAD, I would use ONTO: "The problems started when I downloaded/installed/loaded this new program onto my computer." In summary, INTO is used with things that you type, and ONTO with things that you download or add to the device from a disc.

I would use into. https://www.merriam-webster.com/dictionary/into

Into is better because onto prefer to natural things like surface

I would say: Let me get that information ONTO my computer.