Buying a new router

I have a basic 2Wire router/modem in one right now. I've had this modem and an identical one before it for almost a decade now. It does the job but is inconsistently slow. I've been thinking about getting a new router or find another way to increase my internet speed. I am not very tech savvy when it comes to internet connection type stuff. What are my options? Could I simply connect a new router to my current router/modem combination, buy a new router and separate modem, or buy a Wi-Fi range extender? I have a spending limit of around $200.
Even if I buy a router that is far too advanced for for my monthly service, wouldn't it still make a difference in speed?

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Clarification Request
Is this DSL?

If so it's rarely the modem+router issue but the basic speed of the DSL. And yes you can connect other routers but the step by step is on the web so I'll pause here with a router barely helps on DSL speeds.

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Upgrade my plan

I know you've already dumbed it down and thank you for that, but what you're basically saying is that my only option is to upgrade my monthly service plan? I do currently have DSL. I'd have to get ADSL?

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