@jason_longVery open question but still try to gice some pointers to help.
Assuming you have only 1 website hosted on old server. Recommendation is to move the complete web root folder. For example if website is running under '/var/www' or '/var/html' then copy the whole html or www folder from old server to new server.
You need to migrate the database as well. You have to take care of DB username/password/permissions etc.
There will be bunch of challenges if you are not familiar with Magento. If DB credentials got changed then you need to update the same in Magento.
To test Magento on new server you may need to do the host file entry or change the Magento URL in Magento configuration.
There may be many more thing but assuming you have a simpler Magento installation.
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Magento supports Mysql. You can export from old server and import the DB on new server.
Magento 1 to Magento 2 is rebuild and not the migration. Only data can be migrated but code needs to be rewrite. Recommendation would be just keep what you have without bringing in more complexities of Magento 2.
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@jason_long Yes. Requires tools and bunch on configuration to do the same. I would say only a developer can do it.
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@jason_long On the below URL you can find all the details of data migration tool.
https://devdocs.magento.com/guides/v2.3/migration/migration-tool.html
If you are on latest version of Magento 1 you should be fine. However, Magento will stop supporting Magento after June 2020. You can read about same on below URL.
https://magento.com/blog/magento-news/supporting-magento-1-through-june-2020
Magento is not providing any bug fixes or build more feature on Magento 1. To get support on future release and new feature it would be nicer to migrate to Magento 2.
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