. Suppose the Earths axis had no tilt. Would we still have seasons?

Your intuition is correct. If the Earth's axis was perfectly orthogonal to it's orbit, we would have no seasons.

No tilt, no seasons.  Earth's orbit is an ellipse, so we will be closer to the sun during part of the year.  That would lead to a mini seasonal difference, but nothing like we have now with our 23° tilt.

Precession will give us summer in January in the northern hemisphere in 13,000 years.

We would just BARELY have seasons, because of the elliptical orbit of the earth.  We are at perihelion (nearest to the sun) sometime in January, so that would be the middle of our "Summer," in both the Northern and Southern Hemispheres.  Six months later, at aphelion (the furthest point from the sun), it would be the middle of Winter.

Please note that these "seasons" would be MUCH milder -- in both directions -- than the ones we currently experience because of the tilt of the earth's axis.

No, without the axial tilt, we would not have seasons.  Our seasons are caused by the tilt.  As the Earth revolves around the Sun, the 23.5 degree from the Equator lines are presented to the sun at different times in the orbit.  The hemisphere that is most directly facing the Sun experiences summer, while the hemisphere tilted so that it is less directly facing the Sun experiences winter."The Earth's seasons are not caused by the differences in the distance from the Sun throughout the year (these differences are extremely small). The seasons are the result of the tilt of the Earth's axis."

The summer solstice in the Northern Hemisphere occurs when the Tropic of Cancer, 23.5 degrees north of the Equator, is directly facing the sun (the Sun is directly over the Tropic of Cancer).  That makes it warmer in the Northern Hemisphere.  
At the spring and fall equinoxes, the Sun is directly over the Equator and the hours of daylight and darkness are equal (12 hours each).  
The winter solstice in the Northern Hemisphere is the summer solstice in the Southern Hemisphere.  The Sun is directly over the Tropic of Capricorn.
The Enchanted Learning site shows a good diagram of this.