If your walls and woodwork are in good shape and the current layer is an acrylic or latex paint, then priming isn't necessary as long as you repaint with a good acrylic.
If the same good shape is true but the current layer of paint is an oil base and you want to use an acrylic then priming with an oil base primer is necessary. Acrylic and latex paints have a hard time sticking to oil base paints.
For painting an oil base over oil base, priming isn't necessary. But, a good sanding is necessary to promote adhesion. I like to do the sanding and prime just to be sure.
You will have to use a primer if any repairs are preformed. In this case, use a oil base. Also the new baseboards will also need priming. I like to use oil base primer when ever I can for new wood, but you can use an acrylic primer instead.
The easiest paint to use is an acrylic. It dries quickly and doesn't smell bad. Plus, it is less prone to yellowing than an oil base paint. Check out this page for instructions on <a href="//questions.house-painting-info.com/how-can-you-tell-if-inside-trim-paint-is-oil-base-or-latex.html">how to figure out if your paint is oil base or acrylic</a>.