I spent a long time in Peru in 2011, starting in the south.
As iechegar said, Misti and Chachani are very easy. Apart from the altitude of course. Nevertheless, I enjoyed myself, especially on Misti, because it's simply a beautiful volcano, and a nice layer of snow on top made it even prettier.
More interesting but still a relatively easy climb is Ampato. I had a go at it, but avalanche danger made us turn back. Other relatively easy ones I read about are Hualca Hualca, Sabancaya, Solimani and Coropuna, but I couldn't find partners to go there so I moved on. There is no refugio at any of these.
There are some easy 5000m+ mountains a few hours north of Arequipa on which to get used to the thin air, like Mismi and Huarancante.
In the southeast of Peru lies the Cordillera Vilcanota. Easy mountains are hard to find there, but there is a refugio at Ausangate basecamp. I've heard and read different stories about it, one saying it was private, the other not. However, it's too far to walk there from the nearest village in one day, so you have to bring camping gear anyway. Ausangate is a beautiful climb, and slightly harder than Chopicalqui, and definitely not suitable for people without experience. It's not climbed often, and of those that try it, many fail, not primarily because of the technical difficulties, but because it's a long slog on the summit plateau, which can have deep snow. We had snow shoes, and still it was very tiresome.
I visited Bolivia in August/September 2009.
Like Woodie said, Huayna Potosi has two refugios. I would rate the climb comparable to Cotopaxi. Very likely you'll find a trail. If you want to go there, I suggest you climb slightly lower mountains first to get acclimatized and then skip the lower refugio.
One such lower mountain would be Pequeño Alpamayo. It's a fine climb. A local familiy lives at the lake at the base camp for Condiriri and Pequeño Alpamayo. When I was there in 2009, they were happy to accommodate guests in one of their rooms. I forgot what I paid, but it wasn't much. However, there was no way to ask ahead if there would be room for us, so we still had to bring our own camping gear, just in case. You might find a trail on Pequeño Alpamayo, but that's by no means certain. We had to make our own. Up to Tarija peak, the route is easier than Cotopaxi. From there, it's a bit more difficult, but not all that much. Very exposed though. If you're on Tarija and don't like what you see, you can call it a day from there.
Chacaltaya is another useful acclimatization peak, but it's very easy. Bus loads of tourists are driven up to the end of the road, from where it's half an hour of walking to the top. There is a ski lodge at the end of the road. That may sound rather luxurious, but it's pretty simple. I didn't ask, but I assume you could spend the night there.
Illimani sees a fair number of climbers. No refugio, sometimes a trail. Not when we where there though. The difficulty was between that of Cotopaxi and Chopicalqui. I would hesitate to bring an inexperienced climber there, but if your friend does really well on Huayna Potosi, then you could consider Illimani next.
Ancohuma and Illampu don't see many climbers. No refugio, and don't expect a trail. Ancohuma was a little bit harder than Cotopaxi, but easier than Chopi, Illampu was much more difficult than Chopi. Not an area to go with an inexperienced partner.
Near the Chilean border lies Sajama national park, with a few 6000m+ peaks. No refugio, but you can find a place to stay, and get a simple meal, in Sajama village. By September, there was so little snow left on Parinacota, we got to the summit without crampons or ice axe. Technically very easy, but tiresome because of the scree higher up. There were a lot of people on the mountain, and up to the scree, there was a good trail.
On Sajama we had to descend from high camp because of strong winds - not uncommon in this area. Consequently I don't know first hand what it looks like higher up, on the glacier. From what I read and heard from other climbers, I reckon it's easier than Cotopaxi. We were alone on the mountain, but met some other climbers going up as we went down. There was a sometimes poor trail to high camp on the NW ridge, but it's almost impossible to get lost anyway. Just follow the ridge.