Entering DRC can be a lengthy and frustrating procedure, particularly if you arrive by boat from Brazzaville or by plane in Kinshasa. As well as having your visa in your passport and your yellow fever certificate, it's a good idea to have your original invitation with you, as immigration officers have asked for this in the past. Expect delays, intimidation, being asked to take a seat in a side office for no apparent reason and requests for a bribe, but in general you'll be through within an hour or two. Note that at the Beach Ngobila in Kinshasa, where boats to Brazzaville arrive and depart, it's very useful to engage a fixer who can assist you with the bureaucracy. This can be done at the entrance to the port.
All visitors to DRC need visas, and they're not available on arrival. You must apply at the DRC embassy in your home country or country of residence.
The exact requirements for visas vary from embassy to embassy, but in general you will need proof of hotel booking, yellow fever vaccination and a legalised letter from a sponsor in DRC. Visa fees tend be between US$100 and US$200, and normally require several weeks to be processed.
The only current alternative to getting a visa at home is the two-week, single-entry tourist visa issued for people visiting Parc National des Virunga. The cost of this visa is US$105, and on top of that you'll need to purchase a mountain-gorilla trek permit, a Nyiragongo trek permit or accommodation at the Mikeno Lodge in order to get the paperwork issued. The visa limits you to visiting North Kivu province, and is issued on arrival, though all the bookings need to have been made several weeks in advance. See http://visitvirunga.org for more information.
Visas for Onward Travel
Angola Tourist visas are only issued to residents of DRC. You might be able to get a transit visa but even this was very hard to get at the time of research. Most people end up having to post their passports back to their home country and apply from there.
Central African Republic A one-month, multiple-entry visa costs US$150 and requires two photos and a photocopy of your passport and DRC visa. You can wait three days or pay US$20 for same-day service.
Congo Bring a photo and US$80/120 for a 15-day/three-month visa. They're typically ready in two days, but you can pay an extra US$90/170 for same-day service.
Tanzania Single-entry tourist visas cost US$50, though Americans pay twice that. Multiple entry visas cost US$100. Bring two passport photos and expect it to take 48 hours.
Zambia Single-entry tourist and transit visas cost US$50. Transit visas are easy to come by, but tourist visas are complicated and involve multiple letters of invitation from a host in Zambia.
Extensions are almost never possible and everyone who has attempted to do anything at the DGM offices in Kinshasa has usually regretted it.
Customs checks are not generally that thorough in DRC: it's immigration that's the headache. Once through that, your belongings will be given a cursory look over, but it's unlikely you'll have any problems. There's no need to list electronic equipment or currency you're holding, though officers may ask.