Mutes

Priority order, these don't need to be purchased all at once. As you become more advanced as a player, the literature may require more specialized mutes. 

Straight A metal (aluminum) straight mute is required. Recommended brands: Jo-Ral, Tom Crown, Denis Wick. 

You'll find various metals on the bottom - aluminum, copper, brass - all have slightly different sounds. Start with aluminum, then experiment as necessary. Yamaha and Bach are making metal mutes as well. 

Cup - Denis Wick, TrumCor, Jo-Ral, Humes & Berg. Make sure the cup fits your bell size and that there is complete access to all ranges. 

Harmon - Jo-Ral, Denis Wick. These start to get pretty expensive, but will become necessary for some advanced literature. 

Practice/Whisper - Denis Wick, Best Brass (fits in bell, in case) Yamaha Silent Brass. These are special use mutes only. Do not plan on doing the bulk of your practice in a dorm room with a practice mute. 

Other - Plunger (any store), SoloTone, Pixie, Bucket, hat.

Lubricants/Cleansers

The best way to ensure that your instrument plays its best and lasts for many years, is to keep it clean and lubricated.

Slide -­ Yamaha Slide Lubricant, Slide-O-Mix. Keep your slide clean and lubricated. 

Valves depending on the type of valve (rotor, Thayer etc.) some oils work better others - discuss it with me. Keep your valves clean and lubricated. 

Tuning Slide -­ Leblanc, Dow Corning high vacuum grease.

Other - Cleaning rod (never use the cleaning rod on the inside of the inner slides - use it for the hand-slide only), flexible snake (for inner slide cleaning), mouthpiece brush (keep your mouthpiece clean, a daily rinse is best), soft rag.

Electronics

Electronic devices are changing so rapidly that it is almost impossible to make recommendations that are not obsolete by the time they are posted here. Smart phone technology now makes it possible to have several useful training devices in hand. Stand-alone (Korg, Dr. Beat) devices are still available and work very well.

Recorder - a very valuable tool for improvement - a very unbiased set of "ears." Most phones come with some sort of recording capability. Using a better quality external microphone can often enhance the recorded sound. For much better portable recorded sound quality, WAV/MP3 recorder/players are an alternative. Zoom, Edirol, Olympus, are some brands - built-in microphones, records to flash card, USB to computers, playback through stereo systems or headphones.

Any type of recorder will allow you to listen back for many aspects of playing - time, intonation, articulations, etc. 

Metronome - several excellent apps. Currently I'm using Pro Metronome, Tonal Energy tuner (see below) also has a metronome function. 

Tuner Tonal Energy, multi functional, excellent app. Also Clear Tune. 

External Speaker - several small portable models, I am using a JBL Flip (bluetooth capability). These allow metronomes to be heard while playing, playback of recordings, etc. 

Miscellaneous

Trombone Stand -­ UMI, K&M, Hamilton - I don't recommend the trombone stands that fold up to fit in the bell for storage. They tend to be less secure.

Other - wire music stand, gig bag, locker, Wenger practice room.