How can a 2-sphere exist in Euclidean 3-space? I don't know if this is a simple question to answer however, I have trouble understanding how a spherical object (such as a planet) with positive curvature can exist in Euclidean 3-space with no curvature. From my understanding Euclidean geometry seems to be the most likely description of space as we know it. Is our spherical planet resting atop a flat piece of spacetime or is it surrounded by a flat plane?
 A: I think you are confusing the positive-curvature of the object's surface with the curvature of the ambient space.
We can draw a 2D sphere in 3D Euclidean space. the curvature of the surrounding space would be 0, but the induced curvature on the sphere would be positive.
Just as a comment, a physical massive object does influence the spacetime curvature and makes it non-Euclidean.
A: Curvature is a tricky subject.
There are two misunderstandings in your reasoning.
1 - Curvature doesn't really have a lot to do with the shape of objects: it's about the shape of trajectories. Semplifying: if you start going in a straight line, will you continue going on a straight line? Someone going in a parallel direction will always stay at the same distance to you? If the answer is yes, then you're in flat space.
You'd then say that a manifold (an object) is curved if its trajectories are curved. Again, I'm simplifying, but this is the core concept.
A sphere is a curved manifold if you stay on it, if your feet are sticked to its surface, and you can't move in nor out of the sphere, but just on it. In that case, if you and a friend started on parallel lines, you'd soon meet.
2 - Geometrical Curvature and General Relativity are related, but not equal.
Geometrically, you can have a curved sphere (curved in the sense mentioned above, so if you stay on it) in a flat space: just draw a circle on a piece of paper. If you limit yourself to be moving on the circumference, you will be back where you started, while if you don't set this constraint you can easily go on straight paths. Curved on the sphere, flat elsewhere.
General Relativity tells us that mass (and energy) curves spacetime. This means that the fact that a sphere has mass (not the fact that it's spherical) will curve the (otherwise flat) spacetime around it, and it will be more curved the closes you are to the mass.
P. S. if your

resting atop a flat piece of space-time or is it surrounded by a flat plane?

is due to a mental image like this, see this question and the related answers.
