I've used Fidelity for well over a decade and have been very happy with them.
I do bill paying through them also. I love their online services. Very comprehensive. Plus they have some good retirement planning tools.
I know that they have offered to do a professional evaluation of my account for me, but I have never taken them up on it.
One work of advice - pick a very, very safe password. That means don't use your child or pet's name even with a number added to it. A random combination of caps and lower case, numbers and symbols. Something that would be virtually impossible to guess. It should be something that is very difficult even for you to remember!