Google Photos (formerly Google Picasa) is basic, but works absolutely fine. It's what I use.
They have two storage options when you sign up: "Original Size" (limited storage - you can buy additional storage), which retains your photo's original resolution, and "High Quality" (unlimited free storage), which automatically resizes your photos to a reasonable size (around 800 X 600) which works perfectly fine for most Internet usage, including forum posts. Besides, the forum is set to limit photo size to 800 pixels wide anyway.
The only catch is they don't have a button/form that automatically gives you the IMG tags for posting in forums, but even that is not a big deal. Instead of clicking a button that gives you the IMG link, you right click on the photo and "Copy the Image Location." Then in the forum post click the "Insert Image" button, and paste the URL it inside the IMG tags (in fact this method works for ANY photo hosting site).