Travel from home to the MTC: Any charges for checked baggage are not reimbursed. Recommendation: Check which airlines allow 2 free bags and then ask Missionary Travel (and note your preference on the travel request) to use one of those airlines (for that reason) for travel to the MTC. Or, if you are traveling to Utah before your mission and can drop off bags someplace there, do that to save the checked baggage fee.
Travel from the MTC to your mission: Checked baggage is reimbursed within the guidelines. Currently that is 2 bags, each less than 50lbs, and 1 carry on per person. Beyond the checked baggage limit, fees can be $200 per bag, more if overweight or size, and varies with each airline. Each time you change airlines the fees may be assessed.
Vaccinations: The total cost can mount up. Some vaccinations are required for all missions. Some missions such as some in Africa require many vaccinations. Check with the County Health Department and private sources. County Health does not provide all the vaccinations that may be required. In Arizona the cost of private and county resources is about the same overall. Some are less, some are more with each provider. Vaccinations can cost $1000.00 or more per person if you have to have all of them. The cost may be tax deductible if you meet the IRS medical expense criteria.
You may want to contact the Mission President and office couple early on. They can give specific advice as to what to bring, what not to bring, what is very expensive in the country, what is provided by the mission and what is not, money handling in the mission, cell phones, vehicle rental. Very valuable for planning finance handling and baggage.
Odds & ends: These can mount up. Police clearance letters, fingerprints, certified copies of documents, international drivers permit, driving records, etc, do not cost much individually but add up. Some missions require 23 visa photos. At $4 or $5 a photo, that's a tidy sum. If you can take the photos yourself, size them to the correct specs, and print them yourself, you can do it for a lot less. Just make sure they are acceptable to Missionary Travel.
MTC board and room: Currently $44.50 per night per couple. There is one late night, Tuesday night devotional. You don't have to stay there if you can stay close by and get there each morning in time for class (8 AM).
Medical & Dental: Physical exams may not be an insured cost. Check with your insurance to see what will be paid for. Check with your doctor to see what he/she will charge. The exam requirements may be more or less stringent than the requirements published on lds.org.
Medications: In some countries you will be required to take antibiotics daily starting 2 days before your in-country arrival and for 4 weeks after you leave. The antibiotics are supplied to you at no cost. You may want to plan for the consequences of daily long term antibiotic use. That cost is not reimbursed.
Conflicting information: There may be (will be) conflicting or unclear information. Don't hesitate to call the Missionary Department to clarify. It can make a difference!
Insurance: You will probably get a letter with your call that says your insurance has been reviewed and does not meet the requirements... and a brochure offering insurance for $265 (Sep 2011) per month per person. If you know your insurance is good then call the Missionary Department and insist on a review. It may take more than a few calls and perseverance but if you can get your insurance approved it will save a chunk of money.
Optional medical information: If you are going international, consider the quality and availability of health care in your area, the risk you are willing to take, and how much you are willing to self-insure. Some foreign hospitals require payment in full upfront, or a deposit upfront, or payment in full before a patient is discharged. This could require $50,000 available rather quickly. If you require a medevac the cost can range from several thousand dollars to well over $100,000 for international repatriation depending on medical condition and needs. This is strictly a personal decision but something to be aware of for a mission or any travel. If interested, search the web for the best plans. Read the fine print. Evaluate for criteria, choice of care, cost, pre-existing conditions.
Clothes and other things: Dependent on what you have and where you're going but there will probably be some expenses to meet the mission requirements.