I've done both. To my experience...
DIY - if you have specific places you want to go to, at your own pace, in the order you want to visit, and a bit of budget to have a bit of flexibility. Recommended for those who have been to or likes "arranging things".
If you have accounts on Trip.com, Booking.com, Agoda, Klook, Trainline, Trivago, airline rewards, and similar apps, these might be helpful as well. Get an EU data SIM or e-SIM---helpful to book and check trains, flights, maps, etc.
A&O hostels---present in major EU cities. good rooms, nice location, breakfast simple but always good.
city-sightseeing -- hop-on-hop-off bus tours in major EU cities. Simple, comfortable, affordable
Tour - if you don't want to go through the effort of booking flights, hotels, local transport (train or plane), etc. Tour packages cant tend to feel a bit "rushed" at some point. So choose the tour package carefully. If it's on the low cost end, don't expect too much--check carefully what are included and what's "pay on site". Autumn & Spring presents best weather to travel around EU, but may cost higher. In Summer, specially some smaller hotels may not have A/C, ask if there's a fan at least.
PS.
- Most restaurants in EU serve bottled water for a price.
- Some public WC (Water Closets/toilets) requires a minimal fee (50cents to 1 EUR).
- if you're bringing cash, bring 100 EUR or smaller bills. The 500 EUR bills may not be as easy to use in most places.