You can certainly attach your patch at an angle if you prefer. In this case, you do not need to line the logo up with anything. If your patch is heat-activated, do this on an ironing board.
Let the patch air dry for 3-5 minutes so that the glue has time to cure to the fabric.
Tie a knot at the end of the stitching thread to hold it in place or leave the needle loose. Run the thread just inside the edge of the patch to avoid running thread through the stitching on the edges of the patch. This is less important for appliques, which are usually cut out.
Cut the excess thread off when you’re done. You can use any stitch to attach the patch or applique to the clothing. A running stich is the easiest pattern to use for this.
If there is a sewing kit in your home, there should be a seam ripper in there. You can buy a seam ripper for $3-5. Before you do this, turn your clothing inside out to see if there are stitches on the back. If there aren’t, you cannot use a seam ripper to remove the logo. You may be able to melt the glue holding the logo in place with acetone as described in the final section, though.
This will be kind of time consuming if the logo is large or elaborate. You have to cut through all of the stitches to remove them.
You can also use tweezers to pull the embroidery out.
You may need to peel the logo off if it’s sticking to the fabric. If the logo won’t come off, apply more heat with your iron and try again. Some stronger logos will require several attempts. It can be hard to tell the difference between rubber, vinyl, and plastic. In general, rubber and plastic logos will always reflect light, while some vinyl logos will not. Try ironing the logo and peeling it off before moving on to another solution.
Regular nail polish remover that is less than 100% acetone may work, but pure acetone is much more efficient. You can tell if a logo is vinyl by feeling it. If it’s smooth and feels like it’s resting on top of the clothing, it’s probably vinyl.
