Classical piece recommendations for a novice

It's an impossible task; like asking someone to nominate nice things to eat.

I'd go along the path you're on at the moment; get one of those endless compilation albums, see what you like & then explore those composers' other works, or at least listen to the whole work the excerpt comes from. Sometimes the excerpt isn't the best thing about the work anyway, & sometimes it's a very poor representation of that composer's output. Try & get used to the idea of listening to longer sections of music too. These compilation albums are primarily for people who like their occasional foray into classical music to be in similar-sized chunks to pop songs, ie under 5 minutes. A Beethoven symphony is about 30 mins; Act 1 of Wagner's Gotterdammerung is over 2 hours, but I've no idea why that thought is so daunting to people when they see films at the cinema which are longer, or when novels can take weeks to get through (on & off).

The title of this album couldn't be more wrong - & I'm not sure what Lesley Garrett's doing on there - but it has some other excerpts on it you might like:

https://www.amazon.co.uk/Only-Classical-Album-Youll-Ever/dp/B000026APH/ref=sr_1_1?ie=UTF8&qid=1477658004&sr=8-1&keywords=only+classical+album