I am not a lawyer. Inheritance laws exist at both the federal and state levels. You should seek actual competent advice from an attorney familiar with estate law in your jurisdiction. The following are general comments.
1a. I like the Nolo stuff. Easy to read, reasonably comprehensive.
http://www.nolo.com.
1b. Generally speaking, for each asset, your Dad could:
i. Name your Mom as beneficiary or as TOD (transfer on death),
ii. Change the ownership from him to him and your Mom as joint tenants with rights of survivorship, or
iii. Have a simple will written up that transfers all of his assets to your Mom.
With the first two options, all that needs to be done is to provide a death certificate. With the third option, the assets will have to pass through probate, which takes time, may cost some money, and may be some hassle - it largely depends on the state your Dad lives in.
Taxes are complicated, but in general there are zero tax consequences when one spouse inherits from the other.
2. It sounds like with both of these things you will want a durable general power of attorney. This is a legal document which allows you to take care of your Mom's affairs. The "durable" word means that it remains in effect even if your Mom becomes mentally incompetent.
3. If you do nothing, your parents' assets will pass according to their state's intestacy laws. Basically, these laws provide a default will for people who don't write their own. If they don't like what their state's intestacy laws say, or if they want to avoid relying on them to stay the way they currently are, then your parents each need to write a will saying how they want their assets distributed when they die.
As the above poster notes, there is a rather high federal estate tax exemption currently. Each state has varying estate and inheritance taxes and related exemption amounts. You can research your state's laws and taxes.
It might also be a good idea to get a durable power of attorney for your Dad. When getting one, the person needs to be mentally competent; once they are not then you have to get a conservatorship, which is the same thing but costs a lot more money and takes more time and you have to basically have a hearing and get the court to agree to give it to you.
It would also be smart for you to sit down with your Dad and as tactfully as you can, ask him to at least make a basic list of his assets. Even if it is just "Checking account at Bank A, piece of property in county X, some mutual funds at Fidelity", you'll at least have a good place to start, rather than sorting through his mail or papers to try to figure it out later.
And all of this stuff applies to your Mom too. We all expected my Dad to pass away first and my Mom to live for 30 years after that. My Mom died three years ago and my Dad is still going strong. What I mean to say is don't make assumptions about who is going to die first and who is going to have worse health problems; you might be surprised.