For example, if I take an existing pet training device and make modifications that allow it to be used in a new way (but also for pet training), can I patent the entire device, or just my modifications? My modifications can be removed (except for the attachments) to allow the device to be used as it was originally intended, then easily re-attached for the new use.
In the case of attachment, you can only patent your modifications along with its other uses. Not the entire product (i.e. pet training gadget).
I don't have much background in patents but, for what I know, you can patent the modification you made under your name. It can be listed in original product's patent as a cited patent (if it was a critical modification which the original product owner would like to add). You can even license the original product and create a new along with your modification. You can earn a good profit without harming any law.