Need help with small church system.

I have experience with sound systems (still a bit of a noob though), so I was tasked with figuring out a system for a church of about 20 people. Our sanctuary is about 20 feet x 15 feet. We are thinking of using audio recordings for worship along with our small worship band. We are also talking about doing some podcasting. I think 8-12 channels, and some wall mounted speakers would cover our needs fine. I have not actually set up a system before, so I have been having issues figuring out what I need. We have not set a budget yet, but we want to keep it down in price. What I have been thinking so far, is the 8-12 channel mixer, a pair of powered speakers, and a couple mics. Also, how would I mic an accordion? Thanks for the help.

20' x 15'? I can't imagine having anything live other than acoustic instruments in a space that small. Did you mean 20m x 15m perhaps?

I know it is a small space, which is why I asked for help. At the bear minimum, I need to record audio and play music and videos. I am not sure how to set this up.

I know it is a small space, which is why I asked for help. At the bear minimum, I need to record audio and play music and videos. I am not sure how to set this up.

Don't know if this is too late or not. There are a ton of ways you could go. Even talking the very minimum, you're going to need a budget of $1500-$2000 to do what you're talking about if you're starting from scratch. Below is what I'd recommend buying if you're trying to accomplish what you want the very cheapest possible way without totally sacrificing quality.
That said, I agree that for 15 to 20 people you may not really "need" any of this stuff. You could probably just play your videos on any TV you can scrape up and record decent enough audio with the built in mic on an iPad.
1 small mixer board (Something like a Soundcraft EPM8 at $249, Mackie VLZ4 at $199, etc)
2 Powered Speakers (EV ZLX-12-P are $349 each and sound great. Technically you could get away with 1 speaker in a room your size. In fact, in a room your size you could even opt for a pair of decent powered studio monitors if you really wanted. Something like a pair of JBL LSR308 for $500)
1 microphone, or more, depending on how many people will be singing/speaking at one time. ($100 - if you're only getting one mic, get something half way decent like an Shure SM58 or Sennheiser e835).
1 audio cable to go out from mixer into recording device (1/4" out from mixer to 3.5mm into recording device)
1 audio cable to go out from Video Player into mixer (3.55 out from PC 1/4" into mixer)
1 XLR cable per microphone to connect mics to mixer.
2 cables to connect mixer to speakers (Probably 1/4" depending on what input/output connections your speakers and board have)
1 TV to display your video
For your recording and video playing device, you could use a PC/laptop if you can come up with one. Of you could use a DVD player to play your videos and something else to record your audio (like a basic CD recorder or any digital recording device that takes an audio input)