iTunes is dead, at least if you are running the latest version of macOS on the Apple Mac and there are new ways to access old features. Podcasts now have an app and here is how it works.
The Podcasts app on the Mac is a great way to browse podcasts, find new ones, listen to them and manage your library. It syncs with the Podcasts app on the iPhone, so you can manage your podcasts in macOS and listen to them on the Mac or on the go using your iPhone.
The Podcasts app is a big improvement on the interface and features that used to be in iTunes and it is now easier than ever to browse podcasts and subscribe to them. It is also easier to manage the playing queue and to stream podcasts to other devices.
Here are some favorite features from the new macOS Catalina Podcasts app.
Sync podcasts with your iPhone

If you listen to podcasts, you probably use the Podcasts app on your iPhone. You could keep the Mac and iPhone separate, but most people will find it useful to keep the two in sync. Go to Podcasts > Preferences and select the Advanced tab. Check that Sync subscriptions across devices is selected (clear the checkbox if you don’t want iPhone podcasts on your Mac).
The playlist queue does not appear to be synced, so what plays next can be different on the Mac and iPhone. It is mostly subscription settings that are synced.
Remove played podcasts

Podcast audio files are compressed, more so than music, but even so, they can take up a lot of space on the disk. Some podcasts are an hour or even two, and they use tens of megabytes of disk space. After a while they will add up to gigabytes of disk space, so automatically delete them once you have listened to them.
Stop the next podcast playing

Do you binge-listen to podcasts, playing one episode after another? Maybe it is just me, but it is something I almost never do. After listening to a podcast for an hour or more, I usually don’t want to spend another hour listening to the same podcast. Stop the next episode autmatically playing by going to Preferences > Playback and selecting Stop when an episode ends.
Configure the skip buttons

When playing a podcast, there are buttons at the top of the app either side of the Play button to skip forward or back. They are useful for jumping forward past a section in the podcast you are not interested in or backing up a bit if you didn’t hear what was said so you can listen again.
The amount of time skipped can be selected in Preferences. Select the Playback tab and select a time from the Forward and Back menus. The options are from 10 seconds to a minute.
Auto vs manual podcasts update

Podcasts may release new episodes daily, every few days, once a week or longer. The Podcasts app can check for new episodes automatically, but how often should it check? It can check every hour, but is this necessary? Probably not.
A weekly or daily check is best for most people. However, with a weekly check, it could be several days before a new podcast episode shows up in the app. Let’s say the app checks for new episodes today and your favorite podcast is updated tomorrow, it will not appear in the app until next week with weekly checks. Daily is best.
There is also an option to check for new podcasts manually and this may be preferred. Select File > Refresh Feeds or simply press Command+R to refresh. You can also do this whenever you run out of things to listen to and want to see if there is something new.
Stop podcasts downloading

Podcasts can be downloaded and saved to disk so they can be played offline or they can be streamed, which needs an internet connection. If you listen to podcasts offline, there is an option to automatically download new episodes or all unplayed edisodes.
Warning: The second and third options use a lot of disk space. Suppose you select the All Unplayed option in Preferences > General when you subscribe to a podcast. Every episode will be unplayed, so the app downloads all the previous episodes, which could be hundreds, each using tens of megabytes of disk space. Don’t do it!
Unless you have terabytes of disk space, select Never for this option. You can always manually download episodes you want to store on the disk.
Stream vs download podcasts

Move the mouse over a podcast episode and it can be played or downloaded and played. The play button on the left streams it and so uses no disk space, whereas the cloud button on the right downloads it and saves it to disk. Downloading is only necessary for listening offline, so save the disk space and stream instead.
Read podcast notes

Each podcast episode has a brief text description with it, but sometimes it is short and says little. It may not be sufficient for you to decide whether it is worth listening to. However, a longer description is sometimes, but not always, available.
Select a podcast episode and then click the i button in the top right corner to read the episode notes. There may be links to more information online in the notes and clicking them opens Safari.
Save/download podcasts

Ctrl+click a podcast episode and there are options to download it or save it. What is the difference? Both seem to download the podcast and save it to disk. Select Downloads in the sidebar to see the podcast or play it offline.
If a podcast show is selected, there may be hundreds of previous episodes, but only the most recent ones are listed. There may be as few as two or as many as half a dozen. Below this is a link to show saved podcasts, so it does have some uses.
When subscribing to a new podcast show, you might want to see a list of all previous episodes. You can’t listen to them all, but you might want to save just your favorite episodes from the list. Downloading them seems to have the same effect though, so it’s not clear what the difference is.
Play podcast next or later

Ctrl+click a podcast episode and there are options to Play Next or Play Later. What do these mean? There is a play queue that can be displayed by clicking the icon in the top right corner of the Podcasts window. Play Next adds the episode to the top and Play Later adds it to the bottom. To play the podcast queue, click the first item and episodes play in the order in the queue.

At the right side of each item in the queue is a three-bar icon. Use it to click and drag the podcast up and down the queue to rearrange the playing order.
If you are browsing shows, mouse over the show tile and a three-dots icon appears. Click it and there is a Play Next option, which adds all the episodes to the start of the queue. This probably isn’t what you want, so avoid adding whole shows with Play Next.
Play podcasts elsewhere

The Mac can stream content to other devices, such as the Apple TV, HomePod and so on. It depends what hardware you have on your home network. Click the, whatever the mountain with circles icon is called, and a list of devices you can stream the podcast to is shown. Select one.
Share podcasts and shows

If you discover a great podcast show or episode, you might want to share it with family or friends, or even share it on the web on social media or in a blog post. Ctrl+click an episode and select Share Episode and then select how you want to share it. What gets shared is a web link that opens in Safari, which can then open the Podcasts app.

When browsing shows, mouse over a show tile and then click the three dots button to display a menu. Select Share Programme to share a link to it so that people can add the show to their own Podcasts app.