Why don't we see solar and lunar eclipses often?

No, there is not a solar eclipse whenever we see a new moon. The reason we do not have a solar eclipse at every new moon is mostly due to the angle of Earth's axis (and by extension, the Moon's orbital plane) to the Earth-Sun line. See the diagram below (as requested) for a visual explanation. In the picture, the Sun is to the left. The upper image shows the orbit of the Moon during winter, the lower image shows the possible regions affected by the Moon at different times of the year.

At different times of the year, the Moon's orbit (which is more or less right above the equator $\pm5^{\circ}$) is tilted with respect to a line between the Sun and Earth. Only in the right conditions will you see a solar or lunar eclipse.
Additionally, it is worth noting that (and this is strictly technical) no New Moon that is not a solar eclipse (i.e. the observer is in the umbra) is a "full" New Moon. However, the sliver of light that is "visible" from Earth for these situations is too narrow and/or dim to be discerned by the human eye.

Image taken from http://www.uwgb.edu/dutchs/AstronNotes/HowSolSysWorks.HTM

For more information, see this similar question: Why is a new moon not the same as a solar eclipse?

The thing is, the Moon's orbital plane is slightly tilted (about 5$^\circ$) with respect to Earth's which means from the Earth's perspective that the Moon's motion oscillates around the Sun's trajectory. On most new moons, then, the Moon is either north or south of the Sun and we don't see an eclipse.

For eclipses to happen, new and full moons must occur when the Moon crosses the Ecliptic. Equivalently, the intersection of the Moon and Earth orbital planes have to align with both the Sun-Earth and the Earth-Moon directions. No wonder they don't happen that often!

Edit: I don't know any good 3D simulator, but I found My Solar System (which I got from Mike's answer to this question) good fun.