What would be the best equipment to use (e.g. speakers) to play "minus-one" backing tracks, while I play my trombone live, in venues such as hotels and restaurants?
In a small venue such as a Hotel Lobby or restaurant you will probably not need to mic or amplify your trombone as it will be loud enough on it's own - unless you want or need to send the audio of your performance into a remote location to play through speakers such as inside the restrooms.
But you will need something to play your backing tracks at a volume that is appropriate for the venue and matches the unamplified volume of your instrument. And a portable battery powered boom box is probably not sufficient.
You will need some type of compact portable PA (Public Address) system. There are many different types of portable PA systems but I will assume that as a solo performer you will want to maintain a small footprint (not take up much space) so I will cover a few smaller systems that set up quickly and take up very little space.
If you can afford to spend close to $1000.00 US a good choice would be a line array compact vertical system that has a dedicated input for your MP3 or other portable audio player where you store your backing tracks. These systems will also accommodate a standard microphone and instrument cable.
My favorite is the Bose L1 Compact System. I have many friends who use this system in live performances and they sound great and provide even sound distribution throughout the venue because of the articulated speaker array (each of 6 speakers is angled in a different direction) and because of the sonic characteristics of the line array configuration. It has a built in "sub-woofer" to handle the low end.
All of the portable line array vertical systems break down to fit in a smaller carrying case and can be carried in one hand. Pictured below are the Bose L1 Compact, the similarly priced Fishman Solo Amp SA220 and the lower priced Fender Expo.
Not all to same scale.
Another option would be a powered PA speaker with inputs for your backing tracks stored on your MP3 player or digital recorder or other device. These vary in power from 50 watts up to over 1000 watts and vary in price from under $75 US to over $1000.00.
I personally own a Harbinger Vari V2112 600 Watt 12" Two-Way Powered Loudspeaker that I purchased at Guitar Center for less than $200 on sale. It has more volume than I need for the restaurants I play in and has inputs for 1/8 inch jack such as you would use with an i-pad or i-phone as well as RCA jacks such as would be used with a boom box or portable CD player. Most of these type powered speakers can stand vertically, be laid on their side to project up (wedge shaped) and have a pole socket to permit mounting on a speaker stand. The Harbinger Vari V2112 has all these options and is pictured below.
Another possible option if you don't need as much volume and could benefit from not having to plug in your system - is one of the battery powered portable PA systems. All of the ones pictured below will allow you to input your backing tracks through a portable storage device such as an mp3 player or smart phone and many also have bluetooth wireless connectivity. They are all lightweight and can either be plugged in or powered by batteries so you don't have to mess with cords to trip over. Pictured below are the Roland Cube Street, the Mackie Free Play, the Roland BA 330 and the Alesis TransActive Wireless system (prices starting at $199).
Front and Back of Roland BA330
There are other options as well including small compact systems with two speakers such as the Peavey Escort, Yamaha Stagepas, and the JBL EON206P. But these systems take up more space and take longer to set up than the options mentioned above.
You could also use a guitar amplifier. Some of the digital amplifiers and most of the acoustic amplifiers have a dedicated input for portable audio player (such as mp3) input.
Finally some of the better karaoke players might give you enough volume to accomplish what you want to do. Basically anything that can play your backing tracks at an adequate volume without distortion could potentially meet your needs. Whatever you get, just be sure there is an appropriate input for whatever external source of music your backing tracks will reside on. Look for either RCA input jacks and/or 1/8 inch input socket or bluetooth wireless input capability if your device has bluetooth capability.
Good luck and have fun sharing your music with an audience.
It depends a lot on the size of the rooms, if you're going to play alone or with a band, and how much you're willing to spend. But I would start looking at small compact PA ("public address") systems like this one (this is just an example of the type of equipment, I'm not endorsing or recommending this, or any other specific brand or model) and take it from there.
Try them at a shop and see if they fit your requirements, go for more potent alternatives if necessary.
For a trombone you'll need a microphone that is not too sensitive, otherwise it will saturate easily with the full blown trombone. In principle a good generic instrument mic (like the Sure SM-75) would do the job, but a microphone specifically designed for brass intruments, like the Sure PGA98H has definite advantages, namely, that you don't have worry with mic placement, less ambient noise, etc. (again, these are just examples, not recomendations)
But these are normally condenser type mics, which need phantom power, that most small compact PAs don't have. So you'll also need an external phantom power supply or settle for a dynamic mic.