Definitely something to try, if you can afford it. While you are in Mwanza you may organize your safari to Serengeti ( only 2 hrs drive to the entrance gate). The less expensive excursions to reserves and parks are specialized, of a shorter duration, and worthwhile. If there's one trip that will change your perspective on life, it's an African safari.
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If you have a lot of time on your hands, I definitely recommend a trip to Uganda. It is the most beautiful country with warm and friendly people and I can fully appreciate why people call it the rising star or the pearl of Africa. Because of its nice infrastructure, Uganda is a good country to be a tourist in, and forget all the rumours about it not being safe to travel round Uganda. It's perfectly safe.
Again, Mwanza is a perfect starting point for a short trip to Uganda. From Mwanza you can go boat, which is cheap, but takes an awful lot of time, and is not completely safe. Or you can fly from Mwanza to Entebbe and back (300 dollars pp). In Entebbe take a dalla (don't take the taxi's that charhe 30 dollars) to Kampala. Stay and unwind in Kampala if you're dying for a bit of 'civilization' or continue to visit some of the national parks. In Kampala a good place to stay is the Red Chili Hideaway - a back packer place where they will help you arrange all the trips you like. E.g. River rafting (Uganda has some of the best and safest river rafting in the world with grade 5 rapids - go with Nile River Explorers), A trip to Murchison Falls and it's national park, chimpanzee or gorilla trekking in the jungle. Go to Uganda - I can only recommend it.
(You buy VISA when you arrive at the airport in Uganda. It's no problem and no hazzle - just pay 30 dollars and you're off)
It goes without saying that you must go on a safari when you stay in Tanzania. It's a wonderful experience and Mwanza is a perfect starting place. Off course you should see Serengeti which is more spectactular than I ever dreamed of. Ngorongoro is also worth a visit, but don't forget Lake Manyara, which is a fabulous national park, so much different from the other two. Also Tarangire is a good chioce, but it all depends on what you find more interesting. There are several ways of approaching this. You can hire a car and a driver to take you all the way and show you all the sites or if you know someone with a 4-WD you can go alone, stay in the more inexpensive camps and take the time you need. The last option (which I would prefer) is probably only for people who stay in Mwanza for a longer time and if you're only there for a couple of weeks, you might be better off paying your way out of it through an agency in Mwanza. You can off course fly to Arusha and make all the arrangements here, but I dare say it is not worth it. Furthermore, if you go with a car and a driver from Mwanza, the trip back in car is included in the price, so you really get 2 safaris!
It may sound stupid, but I think a lot of people don't give themselves time enough to simply walk around Mwanza and take in the smells, sound and colours of the city. Go have an ice cream at Salma's cone, go down to the dukas buy the market, check out the numerous tailors in town, buy cheap jeans in the dukas, take a detour into the smaller streets where shoes, clothes, bags, hair, perfume, radios and mouse traps are sold from raggedy dukas. Finish off with a passion fruit juice at Kuleanas or go down to the Kamanga Ferrey and experience the chaos down here and see the beautiful Bischmarck Rock also featured on the shillings.
If you have some time to kill, or just feel like getting away from the city for a couple of hours, then take the boat to Saa Nane Island just outside Mwanza. Saa Nane is a small island with a horrible zoo but fantastic surroundings, and a lot of Tanzanians go there for the weekend to unwind. The zoo has some monkeys, a lion and 2 hyeenas kept in terrible, small cages that would make WHO and Greenpeace go beserk if they saw it. But don't come here for the animals, come here for the peace and quiet and the lovely landscape. The trip by boat is also very nice (takes around 20 minutes) although the engine on the boat is very old and therefore makes a LOT of noice.
You should definitely go to the market when you are in Mwanza. They have EVERYTHING here! Whether you're looking for food, spices, drums, fertility masks, kangas or masaai jewellery. It's so much fun walking around there at this huge market, but watch out for your bag, because it's crowded and you don't exactly blend in when you're an mzungo! Be prepared for some hazzle. Mostly it's friendly and you can avoid a lot of problems with a smile and a few swahili frases, but avoid the kids who try to sell you plastic bags. They're trouble. You can get a lot of really nice Tanzanian crafts here for very little money. Off course they will start with the mzungo-price, but after a bit of bargaining, you will find yourself walking away with a lot of things for very little money.
If you have the time for it you should go to Bwiru in Mwanza and watch the sun set on the most beautiful surroundings of Mwanza - the Dancing Rocks. It' a huge formation of rocks overviewing the green area of Bwiru and the Lake. It may not seem like an exciting thing to do, but trust me, watching the sun set here is one of my most treasured memories from Mwanza. It's stunning.
If you're crazy about national parks/trekking in the jungle, you HAVE to go to Rubondo Island. Having been in both Serengeti, Lake Manyara and Ngorongoro, I still think my trip to Rubondo was my best safari in Tanzania.
You sail to the island in small boats (the boat trip alone is fantastic) - unless you opt for the more expensive choice of flying there - and are dropped on the beach shore where you can either stay in the luxurious tents or in the bandas. Besides a small restaurant and an air strip there is absolutely nothing else but jungle on the island and there's a good chance you'll be the only one there. Besides the miners tourists haven't really discovered Rubondo yet.
You can walk around the jungle on your own, hire a guide for a walking safari, fish, take a boat safari, chimpanzee trekking etc. You can spot elephants, hippos, crocodiles, antilopes, chimpanzees and probably the most fantastic bird life in all of Tanzania. I can't recommend it enough.
Park fees are about 20 dollars pppn, and another 20 for the banda. The tents are more expensive, but here your own shower and toilet, as opposed to the bandas, where you get a bucket of water and a hole in the ground for toilet. But it's quite fine.
Getting there is somewhat more difficult if you're not flying (about 330 dollars pp), but it is doable. All it takes is a bit of patience. You take a bus from Mwanza to Nnkome, where you spent the night at a ranger station. Next morning they will to pick you up in a boat and take you to Rubondo. The bus ticket is around 3000 tz shillings. The boat trip is around 30.000 tz shillings (ask beforehand), but ask if there's a plane leaving with provision for the camp, and you might be able to hitch a free ride.
Make sure to contact a travel agency in Mwanza on beforehand, otherwise the rangers on the island won't know that you're coming and won't pick you up.
If you get the time, you should definitely visit Ukerewe Island. Ukerewe is a big island situated north of Mwanza in the Lake. Some locals refer to it as 'little Zanzibar', and it is truly a place to relax and unwind and watch the beautiful nature. There are wonderful beaches, idyllic small villages, rock formations, jungle/forest, wide open spaces, and green, hilly landscape. Don't expect tour packages, museums, beach volley tournaments and big hotel resorts. There are no tourists on Ukerewe and therefore no tourist industry. If you visit Ukerewe you should go there because of the local people (who are extremely friendly) and because you want to see the tremendous scenery on the island. Also, the 'capital city' of the Island Nansio is worth a visit. If you haven't gone to smaller cities or villages, Nansio will give you an impression of this. It has a nice market and you can get most things there. There are some guest houses, but I must recommend Gallu Beach as the best place to stay down on the beach in Nansio, Good rooms, safe and a nice restaurant. They can also help arrange some trips around the island. DO go to Ukerewe - it's worth an experience.