It should come as no surprise that your work email and phone are already monitored. They can get every email you've sent or received, even if you've deleted it. They can get every text message you've sent or received, the metadata of every phone call sent or received, and so forth.
To a lesser extent, this also applies to your well-known personal phone and email accounts. Law enforcement can get the metadata (which includes text messages) for these things without a warrant. In the above story, the person doing the investigation wasn't law enforcement, but I'm not sure that's a significant barrier if they can pass things onto the Secret Service or something.
The danger here isn't that you used these things to leak, it's that you've used these things to converse with the reporter before you made the decision to leak. That's what happened in the Reality Winner case: she communicated with The Intercept before she allegedly leaked a printed document to them via postal mail. While it wasn't conclusive enough to convict her, the innocent emails certainly put the investigators on her trail.
The path to leaking often starts this way: innocent actions before the decision to leak was made that will come back to haunt the person afterwards. That includes emails. That also includes Google searches. That includes websites you visit (like this one). I'm not sure how to solve this, except that if you've been in contact with The Intercept, and then you decide to leak, send it to anybody but The Intercept.
By the way, the other thing that caught Reality Winner is the records they had of her accessing files and printing them on a printer. Depending where you work, they may have a record of every file you've accessed, every intranet page you visited. Because of the way printers put secret dots on documents, investigators know precisely which printer and time the document leaked to The Intercept was printed.
Photographs suffer the same problem: your camera and phone tag the photographs with GPS coordinates and time the photograph was taken, as well as information about the camera. This accidentally exposed John McAfee's hiding location when Vice took pictures of him a few years ago. Some people leak by taking pictures of the screen -- use a camera without GPS for this (meaning, a really old camera you bought from a pawnshop).
These examples should impress upon you the dangers of not understanding technology. As soon as you do something to evade surveillance you know about, you may get caught by surveillance you don't know about.
If you nonetheless want to continue forward, the next step may be to get a "burner phone". You can get an adequate Android "prepaid" phone for cash at the local Walmart, electronics store, or phone store.
There's some problems with such phones, though. They can often be tracked back to the store that sold them, and the store will have security cameras that record you making the purchase. License plate readers and GPS tracking on your existing phone may also place you at that Walmart.
I don't know how to resolve these problems. Perhaps the best is grow a beard and on the last day of your vacation, color your hair, take a long bike/metro ride (without your existing phone) to a store many miles away and pick up a phone, then shave and change your color back again. I don't know -- there's a good chance any lame attempt you or I might think of has already been experienced by law enforcement, so they are likely ahead of you. Maybe ask your local drug dealer where they get their burner phones, and if they can sell you one. Of course, that just means when they get caught for drug dealing, they can reduce their sentence by giving up the middle class person who bought a phone from them.
Lastly, they may age out old security videos, so simply waiting six months before using the phone might work. That means prepaying for an entire year.
Note that I'm not going to link to examples of cheap burner phones on this page. Web browsers will sometimes prefetch some information from links in a webpage, so simply including links in this page can condemn you as having interest in burner phones. You are already in enough trouble for having visited this web page.
Burner phones have GPS. Newer the technology, like the latest Android LTE phones, have pretty accurate GPS that the police can query (without a warrant). If you take the phone home and turn it on, they'll then be able to trace back the phone to your home. Carrying the phone around with you has the same problem, with the phone's location correlating with your existing phone (which presumably you also carry) or credit card receipts. Rumors are that Petraeus was partly brought down by tracking locations where he used his credit card, namely, matching the hotel he was in with Internet address information.
Older phones that support 3G or even 2G have poorer GPS capabilities. They'll still located you to the nearest cell tower, but not as accurately to your exact location.
A better strategy than a burner phone would be a burner laptop computer used with WiFi. You can get a cheap one for $200 at Amazon.com. My favorite are the 11 inch ones with a full sized keyboard and Windows 10. Better yet, get an older laptop for cash from a pawn shop.
You can install chat apps on this like "Signal Desktop", "Wire Desktop", or "WhatsApp" that will allow you to securely communicate. Or use "Discord", which isn't really encrypted, but it's popular among gamers so therefore less likely to stand out. You can sit in a bar with free WiFi and a USB headset and talk to reporters without having a phone. If the reporter you want to leak to doesn't have those apps (either on their own laptop or phone) then you don't want to talk to them.
Needless to say, don't cross the streams. Don't log onto your normal accounts like Facebook. If you create fake Facebook accounts, don't follow the same things. Better yet, configure your browser to discard all information (especially "cookies") every time you log off, so you can't be tracked. Install ad blockers, or use the "Brave" web browser, to remove even more trackers. A common trick among hackers is to change the "theme" to a red background, as a constant subliminal reminder that you using your dangerous computer, and never to do anything that identifies the real you.
Put tape over the camera. I'm not sure it's a really big danger, but put tape over the camera. If they infect you enough to get your picture, they've also infected you enough to record any audio on your computer. Remember that proper encryption is end-to-end (they can't eavesdrop in transit), but if they hack the ends (your laptop, or the reporter's) they can still record the audio.
Note that when your burner laptop is in "sleep" mode, it can still be talking to the local wifi. Before taking it home, make sure it's off. Go into the settings and configure it so that when the lid is closed, the computer is turned completely off.
It goes without saying: don't use that burner laptop from home. Luckily, free wifi is everyone, so the local cafe, bar, or library can be used.
The next step is to also use a VPN or Tor to mask your Internet address. If there's an active investigation into the reporter, they'll get the metadata, the Internet address of the bar/cafe you are coming from. A good VPN provider or especially Tor will stop this. Remember that these providers increase latency, making phone calls a bit harder, but they are a lot safer.
Remember that Ross Ulbricht (owner of dark website market Silk Road) was caught in a library. They'd traced back his Internet address and grabbed his laptop out of his hands. Having it turn off (off off, not sleep off) when the lid is closed is one way to reduce this risk. Configuring your web browser to flush all cookies and passwords on restart is another. If they catch you in mid conversation with your secret contact, though, they'll at least be able to hear your side of the conversation, and know who you are talking to.
The best measure, though it takes some learning, is "Tails live". It's a Linux distribution preconfigured with Tor and various secure chat apps that'll boot from the USB or SD card. When you turn off the computer, nothing will be saved, so there will be no evidence saved to the disk for investigators to retrieve later.
While we are talking about Tor, it should be noted that many news organizations (NYTimes, Washington Post, The Intercept, etc.) support "SecureDrop" accessed only through Tor for receiving anonymous tips. Burner laptops you use from bars from Tails is the likely your most secure way of doing things.
The point of this post was not to provide a howto guide, but to discuss many of the technological issues involved. In a story about White House people investigating leaks, I'd like to see something in this technological direction. I'd like to know exactly how they were investigating leaks. Certainly, they were investigating all work computers, accounts, and phones. Where they also able to get to non-work computers, accounts, phones? Did they have law enforcement powers? What could they do about burner phones and laptops?
In any case, if you do want a howto guide, the discussion above should put some fear into you how easily you can inadvertently make a mistake.