Some popular devices are:
Fire Stick (Amazon)
Roku
Apple TV (this is pretty upscale)
The first two don't cost much and people sell them used all the time on like Craigslist. A basic one you can get for like $10 used, or $30-40 new.
These plug into an HDMI input on your TV just like the DirectTV cable box does. So switching to the streaming box is a switch of inputs on your TV. Your TV remote can do this. Then you'll have an interface presented by the streaming device, it will have TONS of pre-installed options for streaming services (netflix, amazon prime, HBO, etc), most of which are 'subscription' based there will be some that are free. The streaming device will have it's own remote.
If you have a Smart TV though you'll already have a bunch of these options on the TV, you just have to switch it away from the cable box input to the TV's 'native' state where it's running it's own operating system.
IIRC, Netflix is like $12.99-18.99/month or so, depending if you want it in Standard Definition, High Def, or 4k. Prime Video is included free if you're an Amazon Prime member, which is the main reason I am one of those.
IMHO both these services are well worth it, lots of great shows on both. Some are exclusive to the platform, some are network TV (or paid services like HBO) from year's past ... like Netflix might put the show Seinfeld on for awhile, and you can watch any episode of Seinfeld, on demand ... that sort of thing.
All the big networks (ABC, NBC, Fox, ESPN, etc) have streaming apps that will be on your smartTV or streaming device as well, and often you can avail yourself of these just on the basis
of having a DirectV subscription, no additional fee.
Streaming shows are, almost exclusively 'on-demand', therefore there's no need to DVR. You just pick a show and press play. And yes you can FWD/REW on them.
These are entirely different from cable service (though the DirecTV subscription can help you get some streaming 'channels' for free), and yes, requires the internet of course. The device will have to connect to your wi-fi network that's connected to the web.
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