I'm fairly new at design and I'm currently working on a logo. I want get a certain type of effect. I included a picture of the effect and the logo. I've been trying different effects and filters in both Photoshop and Illustrator but I feel I'm just spinning my wheels. Any starting point would help.
This looks like a Bad Copier effect; instead of sharp edges and thin lines, the edges are blurred, and thin lines have all but disappeared.
I'd use PhotoShop for this. The total effect can not be easily mimicked with Illustrator, and in PhotoShop it's just a few steps.
Your own image is too sharp. Throw in lots of blurring so the thinnest lines all but disappear (you may want to keep them visible just a bit). To get the slight wobble, try a small ripple effect. Finally, adjust the contrast so the lightest parts disappear entirely, the darkest parts revert to black again, and you get a nice sharp-ish edge back.
If you want to regain super-sharp edges, work in grayscale at a much higher resolution than the one of your target. When satisfied with the result, convert the whole image to a monochrome bitmap (if you lose too much or too little image data, either adjust the contrast again or change the black-and-white threshold in the to-bitmap dialog). As a last step, convert it back to gray again and choose a translation back from 2 to 4 pixels, so you regain some antialiasing.
That last step depends on how much larger you made your initial "working" image - for twice as large, choose "2", for thrice, choose "3" and so on. Larger values will soften the gray image more so you'll have to test what works best.
The rough text effect you're looking for can be achieved by using ripple effect in photoshop.
Convert artwork for smart filters > Apply ripple effect under distort.