Quick explanation: (Honestly)
There are two types of format....lossy and lossless. This relates to the compression of the file, JPG for instance is a 'lossy' compression, PNG is 'lossless'.
JPG isn't actually a file format, its JFIF, but to keep things simple the 'jpg' format is very good at compressing images but should only really be done once. The main mistake people often do is to use the 'jpg' file from a camera to work on which, because its a 'jpg' has already been compressed once in the camera. You've not done that so no worries there.
'JPG' is lossy because it removes information from the image. To the naked eye you probably wouldn't notice but you wouldn't want to use one to work with.
'JPG' does not support transparency...period.
PNG is lossless as it retains all the information yet still compresses the file pretty good. If you have any 'transparency' in the image then ideally you should use PNG-24. PNG is based on the old outdated GIF format and stands for 'Png is Not Gif'. Compression is excellent given that its 'lossless' although as with any file how well a file is compressed really depends on the image.
TIFF files can retain layers just as PSD's, are lossless and usually used if you need to pass on a layered file to someone who doesn't use Photoshop.
BMP is also old and outdated although the format is still used in some applications.
WBMP was developed for small devices like mobile phones and is generally unused now....although even CC still has the option to save to it.
GIF will also do transparency but its purely an 'on' / 'off' affair.....no gradients.
As to what format you use really depends on the end use.
Regards.
MrTom.