Here is what to do in Leyte:
1 - Tacloban
Tacloban is an important tourist spot as it holds an important place in Filipino history. A couple kilometers out of town, in Palo, is where Mac Arthur disembarked on October 20, 1944, fulfilling his promise to return. The spot is now marked by the famous Mac Arthur Landing Memorial. A must-stop for every Filipinos visiting Leyte.
Linking, Leyte to Samar, the San Juanico bridge is the longest in the Philippines (2,16 km), nothing special but Filipinos tourists usually don't miss the chance to visit it and take a couple of selfies there.
Another point of interest would be the Santo Niño Shrine and Heritage Museum. This heritage house was built by Imelda Marcos (wife of the dictator Ferdinand Marcos), she never got the chance to sleep in it as it has ceased after the Marcos government got overthrown.
2 - Biliran
Little Biliran is really worth spending a couple of days, the interior is full of terraced rice fields and coconut tree farms. If you want a taste of rural Philippines, Biliran is the right place. Enjoy swimming in one of its many waterfalls, hike to magnificent viewpoints or take a warm bath in a volcanic pool.
Biliran's coast also offers a lot of amazing beaches and strange rock formations, just hop on a boat and enjoy island hopping for the day.
3 - Kalanggaman
Kalanggaman island is famous for its sandbar, white beach, and translucent water. You'll need to arrange the boat with the tourist office in Polompon. It's usually a day tour but you can camp there if you'd like.
Entrance fee: 500 pesos for foreigners, 150 for locals. Boat rental starts at 3000 pesos good for 15 persons.
For more info check out the tourist office website. They have a forum where you can find other people to rent the boat with.
4 - Lake Danao
Lake Danao is a hidden gem only 20 km away from Ormoc. Surrounded by nature, the lake offers a peaceful retreat from the busy city. You can rent a floating nippa hut and enjoy the fresh air, relaxing in the middle of the lake. For the more adventurous type, you can rent a kayak and explore the lake. There are also a couple of waterfalls you can trek to. You'll need to find someone to guide you and be prepared to trek through the forest with no real path to follow.
There's also a river just next to the lake you can swim in and a view deck a couple of kilometers away.
It's mostly frequented by locals so avoid weekend if you don't want the place to be crowded. Getting to Lake Danao can be tricky, there are not a lot of trips there, from Ormoc you can take a jeepney but it gets full really fast. You can also hire a tricycle (500 pesos), which can be better as he can drive you to the viewpoint, and show you the nice spots.
5 - Canigao Island
This is definitively my favorite place in Leyte, first because the island is beautiful and second because it's super easy and super cheap to go there. Contrary to most municipalities, Matalom doesn't ask you to rent the whole boat, they just wait for the boat to be full or almost and they make the trip to the island. It only cost 105 pesos for the boat and the entrance fee, you can spend the night there (tent: 100 pesos, bungalow: 700 pesos) or just do a day trip. The last trip is around 5 pm.
The island is just perfect, white sand, clear water, super clean. No one lives on it, only the staff. You can easily walk around the island in about 15 minutes. On the other side, you can have the beach all to yourself as most people stay around the boat drop off. Bring your own food and water, there's nothing there.
Be careful, the departure point is no longer in Matalom but in Itum.
Near Matalom you can also explore Hitoog Cave and Karap-Agan falls. You'll need to hire a habal-habal or find a motorbike for rent.
Entrance fee: 40 pesos - Boat back and forth: 65 pesos
6 - Limasawa
Limasawa island is rich in history and has a special meaning for Filipinos: Magellan conducted the first Catholic mass of the Philippines on Limasawa on March 31, 1521. This is actually quite controversial as some people claim that the first mass was in fact held in Masau in Mindanao.
Anyway, the island is worth a day trip, you can enjoy some nice beaches and witness a simpler way of life.
If you're in Southern Leyte and you like diving, nearby Sogod Bay is one of the best spots in the Philippines and the perfect place to see wild whale-sharks.
Note: Leyte is not the best place for a solo backpacker on a budget. Most of the nice things to do here are island hopping, and except for Canigao, you'll need to hire the whole boat for yourself, which is quite costly. An alternative would be to go early at the tourism office and try to see if a group is going and ask to join them. It's a long shot, but during weekends you might have a chance.
Note 2: Most people in Leyte have probably never seen a foreigner there, which mean they probably have never seen a woman wearing a bikini or even a one piece in real life. So you might want to wear a shirt and short at the beach if you don't want to feel out of place. Of course, wearing a bikini is ok but locals don't, so they might stare at you a bit.