First of all, if you are a truly a new u/w photographer, you have done an excellent job in both the images you posted. The shark portrait is "in your face", nice and tight. And, even more, the manta ray is well composed. Mantas are usually a very difficult subject to capture properly.
In terms of borders, it is a matter of taste. All of my prints are laminated, borderless and mounted on gatorboard. There are no frames. I like the way the prints seem to float away from the wall in 3D. Also, the images stand on their own with no accenting frames. On the other hand, most of my paintings are framed. So, it is a matter of taste.
For the web display, you need to think of your primary purpose for displaying the shots. Is it for your own pleasure, to share with others, to sell prints and/or to present to editors. Editors, in general, would prefer no frames. And, you might turn off potential buyers simply because they do not like your frame and either can not or will not try to picture the image without the decoration.
Again, your shots are good.
Scott