I went to replace my front brake pads in 2008 base model fit. I did not know all I had to do was release the one bolt and swivel up the bracket. I uninstalled all three bolts and brake fluid started gushing out, I put everything back together and added more brake fluid to the reservoir, pushed on the brakes several times to get the fluid in the system. When I test drove the car, the brakes were barely working and not very much resistance when applying them. Did I screw everything up? Can this be fixed without taking to a shop?

 

Assuming you reassembled the brakes correctly (actually, even if you didn't), you have air in the brake lines, which means the brakes won't work. You need to bleed the brake system to remove the air. There are tutorials here or you can watch videos on Youtube.

Note that you should always bleed the brakes after service. In your situation (fluid coming out of brake line) you could have lots of air throughout the system, especially if you drained the reservoir completely dry. Be prepared to spend a decent amount of time getting all the air out.

First step - don't drive car again until you fix the brakes.
Second step - buy plenty of brake fluid; you can return unopened bottles. If your brakes haven't been bled recently, pick up some PB Blaster or similar to soak the bleed screws so that you don't snap them off.
Third step - watch instructional videos until you understand what you need to do and what tools you need (e.g., on my 2010 Fit the front bleed screws are 10 mm and the back are 8 mm). Bleeding brakes can be done by yourself but is easier with a helper, so try to find videos showing that.
Fourth step - Confirm that you reassembled the brakes correctly and then bleed all brakes in proper order. As an aside, try not to get brake fluid everywhere as it's bad for paint. For my 2010 (GE8) that order is: driver front, passenger front, passenger rear, driver rear. You should confirm for 2008. Make sure you keep adding brake fluid to the reservoir throughout the bleeding process. If you run the reservoir dry you'll have to start over.
Fifth step - test the brakes in a safe, low speed, low traffic environment (e.g., don't jump straight on the highway).

I'm a big advocate of DIY service and repairs and learning by doing, but I also suggest watching videos or reading the service manual before you start disassembly. I've serviced brakes on a bunch of different vehicles in my life and I still start the job by watching Youtube videos to refresh my memory and make sure I have everything I need. Twenty minutes of preparation saves time in the end.