The current answer is yes, it will continue to expand forever. Hubble and some other people discovered that the universe is in fact expanding.
Much more recently it was discovered that not only is the universe expanding, but the rate of expansion is accelerating. This is attributed by most to "Dark-Energy" which is acting as a sort of anti-gravitational force to push things apart, but is highly theoretical.
The expansion of space is a completely mind-boggling concept. Space itself expands without a need to expand into anything else. The Big-Bang is not an explosion in space, but an explosion of space.
The shape is hard to talk about.
The way sometimes we are taught to think about it is like a creature living on what appears to be a two dimensional plane, such as an ant on a large ball. The ant can start at a point and walk all the way around the ball in a 'straight' path and arrive back at its starting point. This is because there is a higher dimension of curvature which it cannot perceive. The universe may be very much this way to us, except we perceive 3 spatial dimensions but know there is curvature on a higher dimension, which is how Einstein describes gravity.