Isla del Sol
Inca legend says that Viracocha, the bearded god who created the universe, emerged from the waters of Lake Titicaca and created the sun at this location.
The life in the island is tranquil, there is no noise, no motorized traffic and no rubbish on the streets (which is rare for Bolivia).
The only conventional way to get to Isla del Sol is by boat. Numerous tour companies operate full and half-day excursions from Copacabana. You can make reservations at the agencies, or just go down to the waterfront around 8:00 or 1:00. It's also possible to catch boats from Yampupata at the northern tip of the peninsula.
If you don't like to book a tour with a company, you can find a one-way boat to Challapampa (25 Bs, ~2 hours) at the waterfront. You may spend a night in Challapampa, walk to Yumani the next morning and return from there to Copacabana by another boat. EDIT feb 2013 - One way boat to Yumani costs 20Bs. If you go on a tour where they drop you off in Challapampa and pick you in Yumani (few hours later) the cost is 35Bs. It is recommended to walk from Challapampa to Yumani (3h).
Recommend leaving your large backpack with a hostel in Copacabana as getting up the inca steps in Yumani is hard work at altitude.
UPDATE June 2013. There are two companies that run boats from the harbour in Copacabana to the island. One goes to the southern part (Yumani) (1.30 hours) and the other to the northern part (Challapampa) (2 hrs). Both leave Copacabana at 8.30 am and 1.30 pm. The morning boat to Yumani might continue to Isla de la Luna, if there are enough passengers, spend 1.5 hours there and then bring the passengers to Yumani (30 bs for the whole trip). Make sure you remind the boatman a few times that you will have 1.5 hours on Isla de la Luna, because he might drive slower (to save fuel) and then allow you only 40 minutes at the island, which is not enough.
The last boat back to Copacabana from Isla del Sol is at 1.30 pm from Challapampa and at 4 pm from Yumani. Of course you should verify this information as schedules change.
From Yampupata local boatmen can take you across to Isla del Sol for 20 bs per person in a rowboat or for 150-200 bs per group of up to 7-8 people in a motorboat. A stop at Isla de la Luna can be negotiated.
UPDATE July 2014: A standard tour of Isla del Sol begins around 8:00-8:30am when you take a boat to the north end of the island. You can get a standard tour by just going down to the waterfront - there will be many people trying to sell you a ticket. It costs 40 Bolivianos for the day of transportation. The boat ride takes about an hour and a half, so if you get motion sick it's probably best to sit on the top level to get more fresh air. The boat will take you to the north end of the island, where you can choose to spend a few hours seeing the sites there and then returning to take a boat to the south end at 1:30pm, or you can see the sites at the north end and then take the 3 hour hike to the south end of the island. At the high altitude in Copacabana, which makes hiking more difficult, if you're not great at hiking you may want to stick with taking the boat to the south end. It's recommended to leave your backpack at your hotel so you don't have to carry the extra weight. It costs 10 Bolivianos to see the north end of the island, and an additional 20 Bolivianos to walk from the north to south end. The boat gets to the south end of the island around 2:30pm, giving you about an hour to grab lunch (the fresh trout it great), walk around, and buy some souvenirs if you want. It costs 5 Bolivianos to enter the south end of the island. The boat leaves from the south end of the island at 3:30pm, and arrives back in Copacabana around 5:00pm.
There are no motor vehicles on Isla del Sol. It might be possible to rent a donkey.
Unless you're spending the night on the island, your itinerary will be basically determined by the boat tour you arrived on. Frankly, the half-day tour is barely worthwhile as it consists only of a brief stop at the southern end and most of your time will be spent on the boat.
A full-day tour will take two hours to get from Copacabana to Cha'llapampa, two and a half hours to see the museum and make a round trip hike to the Rock of the Puma, three hours to take the boat to Isla de la Luna and back to the Inca Steps at the town of Yumani, and two hours for the ride back to Copacabana. You can also choose to hike from the Rock of the Puma back to Yumani (three hours) and catch the boat from there. (Decide on this when buying tickets, as the boat fare to Isla de la Luna adds just a bit to the cost.)
Again, Yumani has the best offerings. Challapampa has some restaurants where you can get soup+trout for about a small 20 Bs. Challapampa could be good for lunch if you are coming with the morning boat and staying overnight on the island. Many places in Copacabana will sell you a lunch box, convinient for daytrippers. The few shops around have limited stocks, -fruits are mostly bananas and apples.
Yumani is growing fast and already has a lot of tourist facilities. There are some basic alojamientos in Challapampa and Challa. The cheapest hospedajes start at about $2usd per night. At those prices expect basic conditions, everything is pretty clean and decent though. Camping should be possible many places.
Isla de la Luna is easiest visited from Yumani, by renting a boat.
Be aware of attacking dogs who wanna bite. In case use your backpack for protection and throw stones.