If the modem supports port-forwarding the information would be in the manual and you would probably be able to get port-forwarding set up.
Does the ISP offer any other modems or modem/routers that you would rent instead? If the ISP does so you may be able to go online and find the applicable manual(s) beforehand to see what features and functions are available. I.e., port-forwarding.
Or you could just go ahead and purchase your own device but that often leaves you at the mercy of the ISP saying any problems are just yours versus the device or devices they provide. And if you own the device any failures are at your expense to replace. Always a trade-off.
If you have a choice in changing your ISP that is good. However you may be going from bad to worse - who knows?
Consider this: keep the modem you have and just purchase a capable router. Meaning the router will support port-forwarding. The modem becomes just a pass through and your router will do all of of the IP addressing, port-forwarding, QoS, parental controls, and so forth.
You will need to do some research and reading with respect to both the modem and router with regards to the necessary configuration. Reading some of the tutorials in this forum and other sources will help. Also check your ISP's forums - you will probably find some instructions and Q&A type information. Check the other (if any) possible ISPs as well.
What you wish to do is basic - the harder part is just getting it all to fit together when your choices and options may be limited.