I'm a big fan of Mozart and Beethoven - symphonies, piano concertos and Mozart's operas, but I'm still new to classical music. I'd like to progress onto another composer of a similar era to Beethoven but not sure which - perhaps Schubert. It's got to be tuneful, powerful and moody. Any suggestions? Thanks

Schubert would be a sound choice. His sonatas and symphonies have a certain parallel to Beethoven - both magisterial. Schubert's impromptus are superb if you like solo piano music.
Also try Hummel, Clementi, Czerny, Cimarosa, Brahms.

As for similarity to Beethoven, there's Brahms (ditto), Schumann, and, to some extent, Berlioz (I recall reading somewhere that Beethoven was one of his major influences), or Bruckner.

If you are a big fan of Mozart, you’ll probably appreciate Joseph Martin Kraus’s works for piano. There is some similar atmosphere… (IMHO...)

Haydn

The classical trinity.

Look at this symphonies and string quartets, as a beginning.

His symphonies too...a mannheimian flavor that is seen in earlier mozartian symphonies.

Haydn and Brahms are the obvious answers. My only complaint about Brahms is that he has only four symphonies. For his fourth I recommend the Kleiber version, for the others... Harnoncourt conducts a decent set.

YOU can't just 'progress away' from me!
I am a god to you!
You shall kneel before me as your MASTER

I shall convert you yet

^Convert - what as in "convert the already converted"? You need to get your plan of campaign organised a little more effectively.