Generally "bigger" pieces of music are named by category. The main categories for instrumental music would be:
Symphony - 4 movements for orchestra
Concerto - 3 movements for solo instrument accompanied by orchestra
Sonata - 3 movements for solo instrument and piano, or solo piano
String Quartet - 4 movements for two violins, viola and cello
Quintet - 4(?!) movements for solo instrument and string quartet
Larger works are made up of separate pieces of music called "movements" which are usually referred to by the number and the speed of the music, like "1st Movement: Allegro". Then you might include the key the music is played in, like C major or A minor. Sometimes, composers or publishers gave additional descriptive names to the works, like "Pastorale".
So a piece is named like "Mozart's Symphony No. 41in C major, "Jupiter", and if you were talking about a single movement you'd add "3rd Movement, Menuetto".
The major vocal works like Operas, Masses, Oratorios and Cantatas, are made up of lots of songs for choir or for soloists which are usually named for their lyrics. So "Sheep May Safely Graze" is from one of Bach's Cantatas, and so is "Jesu Joy of Man's Desiring" - the titles are the lyrics of the song... The only problem being that most of them aren't in English so it's hard to tell!
Some composers, or their publishers, made a catalogue of all the composers pieces and gave them a number called an Opus Number, so if a composer wrote two piano sonatas in E major, one might be Op.3 and one might be Op. 5 and that's another way of telling them apart. Mozart and Bach and some others have their own numbering system, so Sheep May Safely Graze is from the Bach Cantata with the number BWV208, Mozart's 41st Symphony is K.551.
Some of the most famous pieces are just movements from larger works, like the "Hovis Tune" is actually the second movement of Dvorak's Symphony No. 9inE minor, "From the New World".
When you're looking for pieces of music you might enjoy, you could start with composers you already like and look for their contemporaries - so if you like Mozart you could look at Haydn or Beethoven, if you like Bach you could try Handel or Vivaldi. You could also start with an instrument, so look for flute concertos or sonatas. The compilation CDs are a great place to start too! You can get the Classic FM playlists online too if you miss the name of the piece, which I always do 😂