Redwood is a great wood to use on outdoor projects because it contains a lot of Tannin. That same Tannin, will make it slightly more difficult to paint than other materials. The easiest approach would be to use an exterior stain on it, assuming that this bench will be outdoors. The problem with the stain is that it will require more "maintenance", in other words, recoating periodically (depending on the level of exposure to the elements and sun, this could be as often as every year to once every 5 years or so). If you want to paint it, you can, but you really need to use a good stain blocking primer to lock in the tannin. Something like Kilz. The oil based product will work better than the water based, but you can use the water based if you prefer....it is much easier to clean up, etc.
If you will keep this indoors, paint would be a great finish, but keep in mind that it will eventually need to be re-painted if it is kept outdoors. If kept outdoors, water will always get into the pores of the wood, or gaps in the paint, and compromise your finish. Then it starts to peel and lift off of the wood. With a stain, it usually just fades and deteriorates as opposed to peeling. Recoating the paint will probably require even more work than the re-staining. (You will probably have to scrape and sand the wood, and spot prime before you re-paint vs light sanding or cleaning prior to recoating with stain.) Most outdoor furniture that you see made of redwood (think picnic benches, adirondack chairs, etc.) are just stained with a good exterior grade stain. The stain doesn't last as long, but the redwood has superior resistance to deteriorating due to the elements because of the Tannin, so the wood isn't affected as much. The stain is a lot less work!
Good luck on the Eagle project!