Marci,
I haven't "cut the cord" yet, but I've given the idea much thought of late and I'll pass along some general advice. You did leave out a few details, but I can assume you have HD televisions and you're probably thinking of connecting everything wirelessly.

Start small! This will allow you to dip your toes in the stream (pun intended) before you jump in. The ultimate streaming device would be a small form factor PC, which can play any local content (including DVD/BluRay) along with stream anything you can find in HiDef, but cost adds up quickly if you have more than one viewing location. The second option are one of the myriad of set top boxes being sold to stream content to the TV, but their are dozens to choose from and not all of them are created equal in their abilities to deliver the content you want at the quality you want.

Things are different when all of the media you consume is streamed. You can't count on any one service to have everything you want to watch. It's just the nature of the beast - everyone wants their own piece of the pie, or wants to micromanage their content. So, while you're thinking "drop cable and stream netflix" the actuality is drop cable and stream, Netfilx, ESPN, HBO2Go, buy some shows on Amazon, etc.

What I would suggest is starting small and experimenting before cutting the cord. This will allow you to see if streaming will satisfy your entertainment needs. Probably the best solution would be something like a Roku 3

or a Roku 2 XD
if you have older or non HD TV's. Roku's are popular and versatile so if you find streaming works for you would could buy additional units and have a similar setup for each TV.

If you have an Android smartphone, or an old one you don't use any longer, you could also try a Google Chromecast

.

Another thing to consider is connecting the devices. Wired will always be better than wireless. With the proper signal strength wireless will work, but you could run into problems depending on how many devices will be connected and how crowded the wifi spectrum is near your home. If you later decide that wireless isn't cutting it and stringing Cat5e isn't feasible you might look into powerline adapters like the Linksys Powerline 4-Port Network Adapter Kit

.

As for your DVR, well it isn't a rosy picture. If you own the DVR you may be able to use it for a while (without service) to watch your shows. These boxes use a proprietary video format so you can't just "copy" the files (if you can even read them) and play them from a computer. At best the only option I know of would be to play the shows out in analog formats (either component-best or composite-worst). The only way I know of to accomplish that is with a DVD recorder with component inputs (red/green/blue and red/white for audio) or a Diamond Multimedia USB Video Capture Device

.

Selahgal,
There's nothing preventing you from running powerline adapters like those linked to above.

You wouldn't use the outputs of the TV, but the outputs of the DVR (red/green/blue for video and red/white for audio, or in a worst case scenario composite which is yellow for video and red/white for audio).