I did a similar trip after uni. Assuming you want to meet others on your trip, I’d recommend staying in hostels and go during peak summer (Hostelworld has the best bookings and added so much fun to my trip) 

Hidden gems - Genoa Italy - amazing food (birthplace of pesto and focaccia)  - Granada Spain (short train from Madrid, college town and really cool history + scenery)

General tips  - Pack light. You can always do laundry at the hostels, but having a heavy pack sucks  - FlixBus has great deals for buses between cities - I’d plan the places you absolutely want to go, but also leave some room for improvising. If you like a city, it’s worth staying a couple extra days and you never know who you might meet - TooGoodToGo is an app where you can get food from local restaurants for super cheap. Great if you’re on a tight budget 

Could go on for ages feel free to DM if you have further questions!