If you go straight in the universe, will you go round in a circle and end up in the same place that you started?

Well I'm going to go so far as to say it depends on your definition of 'go' and for that matter, 'place'.

I think the question is meant to be about the large scale geometry of the universe.  So for instance, the surface of Earth is a non-Euclidean 2D surface, and if it was perfectly spherical, you can start anywhere, draw a straight line in any direction on it that never bends left or right, and the line must come back to its starting place.  So is the universe like that?  It could be, but is generally not considered to be so.

Now about the definition of 'go':  If I walk in a straight line on Earth, I will eventually come back to my starting place.  But if Earth is expanding by 1% each day, I simple do not walk fast enough to ever get to that point, but if I had a vehicle that can 'go' fast enough, yes, I'd complete the circuit.  So the universe has a speed limit, and so no matter how fast you 'go', you're not going to get beyond the event horizon, which is currently about 15 billion light years away.  The universe is bigger than that, so not even light can make one circuit.  You can't see Earth if you 'look far enough', even if spacetime was closed in a loop.

About the local spacetime curvature of which chiralSPO is speaking.  Back to the Earth example, except it isn't a perfect sphere, but has smooth hills and valleys here and there.  Now a typical line drawn, never bending locally left or right, will not come back to the starting point except by low probability.  The curves in the surface will bend the path left or right, and you'll end up somewhere else after one circumference worth of line drawing. For the same reason, light going anywhere in the universe is deflected this way and that a little bit every time some large mass is passed.

There are places in the universe bent sufficiently that light will come back to its starting point in a short time, but nothing like that in our solar system.