Small set up (100 people)
FOH Speakers - HK Audio E210 Sub As (Active sub woofer with internal crossover, made up of 2 10" cones) HK Audio Elements E435 x2 (150w RMS active speaker made up of 4 3.5" Cones)
Both are very portable.
Monitor Speakers - SubZero SZFP0212 x3 (These monitors can double as FOH speakers if needed)
Mixing Desk + Effect Units - Mackie ProFX16v2 16-Channel Professional Effects Mixer (this mixer had effects built in)
Microphones + DI boxes - SM57A beta x2, SM58A beta x2, Shure Drumkit mic set, BSS AR-133 Active DI Box x2
Power - 20 kettle leads, 5 extension reels, 5 extension cables.
Cables - SubZero 20 Channel Stage Box, 40 XLR cables, 5 XLR gender changers, 4 1/4 Jack Cables, 5 XLR to 1/4 jack converters.
Medium set up (300-500 people)
Cables - Soundcraft MSB-32R Mini Stagebox 32, 70 XLR cables, 10 gender changers, 10 1/4 Jack Cables, 10 XLR to 1/4 Jack converters.
Large set up (1000+)
FOH Speakers - JBL PRX418S x4, HK Audio L5 115 F x4
Monitor Speakers - Shure P3T Transmitter + 5 P3RA Receivers & SE215 In-Ear Monitors
Mixing Desk + Effect Units - SSL Live. L300
Microphones + DI boxes - Audix DP Elite 8, Audio-Technica AT4033/CL, Telefunken M80
Power - Behringer iNuke NU4-6000 Power Amplifier x 7, 40 Kettle leads
Cables - SB 32.24 Stagebox (This stagebox offers easier connectivity to digital mixers)
Speakers:
Active speakers are good for smaller venues. Their built in amplifiers mean they don't require external amplification. This makes the signal flow to them simpler and also easier to transport as you don't need amplifiers. Crossovers are also built into the speakers.
Passive speakers require external amplification and sometimes an external crossover system. You also have to pay more attention to the power require by the speaker. However Passive speakers are much lighter than Active speakers which is good when you need a lot as you would in a large venue.
Monitors
Stage wedges are angled speakers used give musicians individual mixes on stage. These are usually very directional so if placed correctly are only feeding the ear of the musician and make it easier to control feedback. They are great for small to medium size venues but the larger the stage and the more movement of the musicians the more monitors you will need.
Side fills are large speakers placed at either side of the stage and feed all musicians one mix. These add to stage noise and can create more problems in the way of feedback. Musicians can't have individual mixes.
In ear monitoring removes any worry of monitor feed back and musicians can hear the mix they want to hear perfectly so are great for large venues. However they isolate musicians from each other and the audience. They also reduce the volume on stage.
Mixing Desks
Analog mixers are great for small and medium sized venues. From 4 channels to 24 most of them come with Gain, EQ, Aux sends and Pan controls. The signal flow through these mixers is relatively easy to understand. However any effects such as reverb, delay and compression require outboard rack gear and extra cabling.
Digital mixing desks are great for large venues as they use less space compared to a large analog desk that would be needed to do the same job. They are also great for performances with a lot going on such as band change overs and level changes/effect adjustments between songs. These usually have built in effects meaning there is no need for out board effects (these can still be used if wanted). However the signal flow through digital desks is harder to under understand and visualise.
Microphones + DI Boxes
Dynamic microphones are the standard for live sound. They are more durable compared to Condenser microphones and don't require phantom power.
Wired microphones are great if musicians aren't moving around much. But in medium sized and large venues where musicians move around more wireless microphones are more suitable because cables wont come lose and can't be tripped over.
Clip on microphones are very easy to set up and make stages look much cleaner.
DI boxes are great for blending acoustic guitars with non acoustic instruments.
(1) - https://www.bhphotovideo.com/explora/audio/buying-guide/basics-live-sound
(2) - https://www.soundonsound.com/techniques/using-mics-di-boxes-stage