Welcome to the post-cable future. Come on in, the water's fine!
I'll share my experiences (I cut the cable in April, 2009), in hopes that what I've learned along the way will benefit you.
First, I know little about what media you like, what platforms you like (Win/Apple/Linux/iOS/Android), or how techie you are. So, if I'm dumbing things down too much, sorry about that!
I recommend considering the next step to be not just about a "getting DVR" but about analyzing what media you want and where you want to view it. If all you want is just an appliance box that records via an antenna, check to see what TiVo has at the moment. I used to have a TiVo that did that, long ago (2005?).
When I left cable, I really wanted to improve the viewing experience, getting as much in HD as possible, but now pretty much everything other than DVDs and "old content rebroadcast" are HD, so no worry there.
I recommend a product that really changed my viewing life, called Emby. Check this out:
https://emby.media/
Emby consists of two parts:
1. A server on one of your PCs (or various other platforms) that hosts your media, and which performs the recording of off the air (OTA) programs.
2. Clients that play back the media. There are PC clients (Emby Theater), Android and iOS clients, Roku clients... and so on. Pretty much any recent smart device has an Emby app.
In my case, I have a dedicated PC that acts as the Emby Server and also runs Emby Theater. This PC is near my main TV; the TV acts as the monitor for that computer. Emby Theater is always up; I rarely minimize it to do something else on that computer. So, any of my stored media, including "DVRed" TV shows and other media I've ripped to the server are instantly accessible.
That TV happens to be a smart TV, so I can use Netflix, Hulu, Amazon Prime to stream, as well, needing nothing at all from the PC.
How does the PC record shows? I use a network-attached tuner, from SiliconDust. Your antenna plugs into the tuner. I've got their Extend model, which now has been superseded by the "Connect" model. Both are dual-tuner models so you can record two things at once or watch one live program and record another at the same time. These are all in the "HD HomeRun" product line. I think they are about to release a 4-tuner variant.
SiliconDust HDHomeRun has their own DVR software; I haven't used it. I believe it to be relatively primitive.
Anyway, Emby can be configured to use TV schedules from SchedulesDirect, so it knows what's on, and you just tell it what to record, and it talks to the tuners and records your shows. You then watch the content on any device at home, or anywhere you've got a smart device connected into your server. It's pretty slick.
What does this all cost?
I'll assume you have a PC running some fairly recent (Win7+) Windows hanging around. The server DOESN'T have to be super fast or the latest tech. I was running a dual processor Athlon from circa 2009 until last Christmas when I built a new server. You will want some hard drives in that thing... a Terabyte or two. To start.
So, if you already have a PC, the SiliconDust Connect will run you about $100. Emby Premiere costs $5 per month, OR $54 per year OR $119 lifetime. I am a lifetime subscriber. SchedulesDirect costs about $26/year after a 7 day free trial (you can buy 2 months for I think $6).
If you want to try this out, I recommend buying a one month Emby Premiere license and signing up for the schedules direct trial AFTER you get the Connect hooked up.
I use Emby for much more than DVRing TV. All of my bluray content is ripped in glorious 1080p and stored on the server. I also rent blurays from Netflix and 3d-BlurayRental.com, and TEMPORARILY rip it to my hard drive for viewing, after which I delete it. Note, I'm a big anti-piracy advocate. I pay for all content I watch.
With this setup, there's always something on, I can watch it anywhere, it's easy to record my shows, and the main inteface on the TV (Emby Theater via PC for me), is very simple for anyone to use.
Please let me know any questions; this is my "other hobby" besides rockets.
Good luck!
Marc