Something of a speciality of mine, having done a thesis on him *cough* *cough* years ago.
No 1 - a brilliant student work but there is little 'typical' Shostakovich here.
No 2 - written for the 10th anniversary of the 1917 revolution. A short 20-minute work in one movement in several sections. Not a great work but with some very interesting hyper-contrapuntal writing at the beginning, signifying the choas of the revolution.
No 3 - doesn't really hold together. Hardly ever performed and one of Shostakovich's weakest works.
No 4 - a hard nut to crack and of gargantuan proportions, this is, however, a masterpiece that rewards repeated listening. The première was delayed from 1936 until 1961 due to Party pressures. VERY toothy.
No 5 - the best-known of the 15. A good piece but a big step back in style from the 'dangerous' 4th.
No 6 - an oddity but a wonderful piece. Worth checking out.
No 7 (Leningrad) - has had a poor press over the years and does have its moments of bombast but still worth investigating if you like an exciting ride.
No 8 - Long, tragic and profound. Not an easy one to love but a truly moving experience. One of his best IMHO. Pretty toothy, I'd say.
No 9 - A breath of light relief after the two previous wartime epics. A lovely piece with a lightheartedness (albeit sometimes ironic) you won't find elsewhere in the symphonies.
No 10 - considered by many to be the peak of the cycle of symphonies. Powerful, majestic, contrasted - get this one if you don't already have it.
No 11 - A programatic symphony based on the abortive 1905 uprising in Russia. Rather film-music in style but plenty of toothy moments.
No 12 - a real let-down. Shostakovich going through the motions. I wonder if it was deliberate that the piece dedicated to the 1917 October revolution turned out to be shallow, hollow and bombastic?
No 13 - half symphony, half cantata, a setting of controverisl poems by Yevtushenko. Similar in form to No 8 and very toothy.
No 14 - not really a symphony at all but a song-cycle of 11 movements on the subject of death for bass, soprano and a small chamber orchestra of a few strings and percussion. Dark, profound and harrowing, this is one of the best.
No 15 - an enigma, full of quotations and quirky ideas but some wonderful music once you get used to its odd format.
Also recommended:
The Execution of Stepan Razin - a wonderful, powerful cantata which is a thrilling listen.
Quartets 3, 8, 10, 13 - much more intimate expression. Try these four first and then explore the others.
Piano Quintet - an excellent chamber work - more like a symphony for 5 players.
Lady Macbeth from Mtsensk/Katerina Izmailova - the opera that got Shostakovich into so much trouble in 1936 - and got the 4th symphony pulled from rehearsals. One of his greatest and most graphic masterpieces.
Ballet suites 1-4 - sample the lighter side of Shostakovich. Not great music but entertaining stuff.
Hamlet - music for the 1964 film (not to be confused with the earlier music from a film of the same name from 1932). One of the greatest film scores ever written.
There - that should keep you out of mischief for a few months!