I have nothing and $3000 to spend.

T5i – for the hdr capability and decent video (950), a decent tripod with a flud head (200), an external microphone like a RODE (149), a few softbox lights with stands from cowboy studio (100), an led light for the camera (50).

These are approximate Amazon prices.  All total about 1500.

Save the rest untl you get to know your equipment and you really know whatelse you need.

Erik nailed it. I would suggest the same equipment. That's a legit DSLR setup for under $2k. 

Also consider your options for editing software. If you've got a Mac, iMovie is a pretty free solution. Then there's Final Cut X ($300), Adobe Premiere Pro (about $30 a month) and something like Sony Vegas (about $100).

All have their pros and cons. It really boils down to personal preference.

(PRO TIP: Try all the "free trials" and see which you like best) 

Without sounding biased, I'd recommend maybe working with iMovie, getting comfortable with the interface, and then upgrading to Final Cut Pro X when you need more editing control in your videos. 

I certainly agree with Erik also, although I'm a Panasonic man and the GH3 is a good consideration at about the same price.  However, the very FIRST thing you need is a VISION.  A script or at least an outline.  The better and more detailed your vision, the better you can buy equipment to make it come true.  Write down the storyline and types of shots.  Will you want a dolly?  Will you want a crane?  Will you be doing interviews so you want to add a lavalier mic (not replace the Rode, but add to it).  How will you distribute?  Via website, YouTube, DVD, live streaming of events?  Don't start your thinking with hardware, start with concepts, scripts, types of shots, methods of distribution.  Will you need a new computer for editing and/or streaming?  How do you want to capture, transfer, backup?  Certainly you want a camera to capture HD, but will you distribute SD DVD's or stream SD?   Make sure you include backup drive(s) to avoid losing your archives.  How much experience do you have?  If very little, then get local help and advice.  Will you want 2 cameras and tripods?  A board?  That depends on your vision. Look around at what other churches have done.  Many now stream services and have very elaborate equipment and several people to operate and edit.  Others record, create DVD's and distribute to those who can't attend services.  Obviously, esp with this budget, you have to start small, but have a vision for the future and growth.  Here's where you are – where do you want to be – how are you going to get there?   The principles are the same, whether for a church, a business, making a feature film, commercial or a documentary.  Who are your "customers" and how do you keep their interest, entertain or inform them?  What is your objective?  To win them over to "your way," or simply inform, or entertain?  Most productions have some of all these, but you should have a plan.  Is this simply to complement services and archive, or is to grow the congregation?  In your planning, include how much time of your life can/will you devote to this?  Are there others who will help?  Do you watch TV?  Look at the credits at the end of even a short show (and not a major popular program) and see how many people are involved in a 30 minute show.  There are many one-person production shops, but most are paid or trying to make a profit; and they get help for large productions.  Usually for a church it is volunteer work.  Plan for concept, time, to avoid burn-out, and how to deliver a dynamite message to your target audience and accomplish your purpose.  You'll probably find many locally willing to help, but you have to find them and ask.  Good luck.

Oh my gosh, so much good advice! I agree with all of it. Scubajam mentioned a computer. This is a critical component. The good footage that you take won't do you much good if you don't have a good computer that can handle editing it. This is where your budget might be limiting if you don't already have a good computer. Getting a cheap system from Wal-Mart isn't going to cut it either. You'll need to spend some money to get a decent editing system. I would say quad-core Haswell processor, at least 8 GB of RAM, but 16 would be better, high-end GeForce video card can get you by, but a Quadro would be better. If you're going with a Mac, you're probably looking at even more money. 

Just a little story about finding help locally.  A few years ago my daughter was getting married and of course I wanted to do the production, but I wanted to do nothing video on that day.  I wanted to edit after the event, but needed camera operators who could work with a minimum of supervision.  I put an ad in Craigslist and said I'd pay $150 each for 2 camera operators.  I was flabbergasted!! Within 24 hrs I had over 40 responses and they were all wayyyy overqualified.  Almost all had experience on major TV or movie productions, one worked on Seinfeld.  It seems people in this business work a gig, then it ends, and they're out of work.  But it's "in their blood."  They prefer to work for $$, but just want to be in the loop to be doing something in the business.  They offer advice, help if the time is available.  Look at what you've received here and what it cost.  Seek out local help.  I ended up paying $175 ea (they had not asked for more) and felt embarrassed they were so qualified.  And they did great, much better than I would have done alone.  The shots were all well exposed, in focus, and many artistic.  Seek and ye shall find.  Not during the shooting, but during the editing you'll find out very fast if this is "in your blood."  People are so used to seeing high quality work on TV and in movies, that this becomes the standard by which all video is judged.  It is extraordinarily difficult to put together a well-crafted, stable, entertaining piece that holds interest.  Today's low priced cameras and equipment can meet image expectations, but the real quality is within you.  You'll find out fast if it's in your blood.  again – good luck.

Well said, scubajam.