Switching to an SQL backend should definitely help speed things up. I've personally found a large performance gain even when my database was only 5MB. And the effect becomes greater as the size of the database grows. There are a few options to migrate from an Access backend to an MS SQL backend. Before you start though, make sure all your tables have a primary key and back up your database.
The most obvious option is built right into Access, known as the Upsizing Wizard. It's probably also one of the easier options. You can easily find tutorials on it on Microsoft's website, or you can just try it out yourself.
Another option many people prefer is using SQL server to import. For this, you need to have SQL server installed as well as SQL Server Management Studio (SSMS). Create a database using SSMS, then import the Access database into the new database. You will have to reconnect the tables to your Access application manually. But, the database conversion on SQL server is usually better than what you get from Access.
If you decide to opt for another database rather than MS SQL, for example MySQL, conversion might be a bit more difficult. One of the easier options to do without any programming experience is to export your Access tables into another format that can be brought into MySQL (or whatever database you choose). For example, you might choose to export your tables as text or Excel tables, then import them into MySQL. You will have to redo a lot of the table structure though, such as primary key, data types and relationships. Although, even if you use MS SQL, you should still double-check those things migrated correctly as it still doesn't work the same way as Access.
As for the programming, all of the code should be in the frontend. Therefore, you may not have to change much if anything at all. Personally, I have only had to add a dbSeeChanges option to DAO recordset creation.
