The only train that would need booking in advance is the Eurostar out of London. It has a passholder quota and trains definitely sell out days or weeks in advance. There is no good alternative apart from ferries or buses (full travel day).

Apart from that one I don't think you'll face issues except around Easter where loads of people will be travelling. Around that time either book in advance, stay in a country without mandatory reservations or use reservation-free regional trains.

Most trains require a reservation in Spain and they do not run frequently compared to Italy where there's a hourly service at least on most lines.

The Eurail Pass will give you a lot of freedom. :)

To add on @thibcabe ‘s good advice, since south of Europe is a bit trickier to navigate:

Spain is very reservation heavy. Also expect very busy periods around the Semana Santa, especially in Spain (24th March - 30th March). If planning to be in Spain around that moment, it’s wise to reserve in advance too, for the rest it’ll be okay. Shouldn’t be that busy.

France has a lot of reservation heavy (TGV) trains. But you almost always can get around by reservation free regional trains (TER) as well, often leading you to much more interesting railway lines than the very efficiënt and quick, but rather boring high-speed lines.
TERs are mostly not that frequent, but in generally they have some of the most comfortable regional trains in Europe. If you have time, and want to get of the trodden paths, very recommended to do France “en TER” ;)

The rest of Europe will be much easier to navigate around, with a few handy tricks needed sometimes to save a bit of money or avoid weird supplements. Just ask on the community if you can’t figure it out.