Don't feel old anxious17 label yourself open to new experiences ! I don't have a clue about podcasts either - I just sort of click on them and some play and some don't - looking forward to reading the "proper" way of doing things here though!
On my iPhone there is an app, and you can search by category and see the most popular podcasts. Then you either stream them or download them and it tells you when new ones in the series come out. I imagine it's similar for android.
I recommend my dad wrote a porno, it's hilarious and brilliant. For other people's recommendations maybe just try googling, then you should be able to find what is popular.
There are a few threads in the Radio Addicts topic about podcasts so may be worth a look there.
You need a podcast app for Android. I use iPhone/iPad and on that app you search for the podcast, then toggle a Subscribe button and all new episodes appear in your Library and once you’re subscribed you can access all old episodes easily. There are settings you can adjust. So for mine I have downloads off, newest episode at the top of the list, delete played episodes, etc.
First, you need an app. Some good ones are Pocket Casts, Podcast Addict, or Google Play Music.
Open the app, and find a podcast you like. All good podcast apps should have some store-type function where you can browse podcasts. There should also be a search function to find a particular podcast, e.g. Radio 4's Friday Night Comedy.
Subscribe to the podcasts you like, so you get all the new episodes. It should also let you see past episodes too. Careful that you don't allow the app to download new episodes on your mobile data, it could rack up hunderds of MBs in data usage otherwise.
This is new to me too as I've recently started my own podcast but I too found the listening to others quite confusing at first. If you know what podcast you're after most of them will have a website so you can listen directly via that when you search for it online. Or often they will have their own twitter/facebook/instagram accounts too with links to their websites or iTunes pages etc in the bio.
If you want podcast inspiration and will enjoy searching for them - like has been said Apple Podcasts/iTunes is your best bet if you have an iPhone and also to download onto any computer (doesn't have to be a mac) and there are other podcast players/apps that play podcasts including Spotify, Googe Play and Pocket Casts.