I think it's hard to give general advice for all EU countries. E.g. in Sweden you'd want to avoid the big old banks and choose a ”niche bank” such as Avanza or Nordnet. Then you’d put your savings into a ”kapitalförsäkring” or ”investeringssparkonto”, which are favorably taxed. However, this option is only available for Swedish citizens. So what’s best for you in terms of fees and taxes will probably depend highly on which specific country you live in.