I usually steam my carrots, potatoes, and green beans before adding them to almost done meats (as opposed to cooking them all together from the beginning).
I have hit on a tip for potatoes in the past year or two that has made potato prep so much easier. This tip works for potatoes you want to peel or those you do not wish to peel (in that it helps clean them too). You should, of course, cut out bad spots before doing anything with the potatoes.
1. Put whole potatoes in large pot of boiling water.
2. Cook potatoes to the doneness you desire, boiling vigorously. (This will depend on the use you have for them. If you want them partially cooked or you just want the peeling off easily, etc.)
3. Drain the potatoes (use one side of the sink for draining and the other for peeling, so the heat from the drained potatoes and water doesn’t make the sink too hot to do the next step).
4. Plunge the drained potatoes into the other side of the sink that is filled with ice water.
5. In five minutes or so, simply reach into the ice water and start rubbing the skins off (or do not leave in water long, but get them out and use with peelings on). The skins rub off with your fingers only and very quickly.
6. Now cut, grate, slice, dice, julienne, or leave whole for whatever you are making.