Child support does not change upon remarriage per se, although remarriage often results in a change of circumstances that does require a change in child support.
Alimony often terminates upon the remarriage of the person receiving it, but not always. Usually, it does not terminate upon remarriage only when a separation agreement reached by the parties says that it doesn't terminate.
Thus, generally speaking, a man's child support and alimony obligations will not be reduced because he has a new spouse and/or her children in his household.
Child support does not change based upon marital status. It is a function of the income (of all types) of each of the parents, of extraordinary expenses of the child, and of overnights spent with each parent per year (it can also change if a child, perhaps one of several provided for in the child support award, is emancipated).
Special rules apply when child support is owed to more than one co-parent (e.g. someone with two ex-wives and five children from a combination of each of the prior spouses and a current spouse).
Critically, child support is not a function of the expenses of the parents or of the non-extraordinary expenses of the children.
Income for child support purposes does not distinguish between earned income and investment income, or between taxed and untaxed income, and has a very broad definition of income that sometimes includes untaxed gifts. But, as a matter of policy, some child support laws disregard second or third jobs, or earned income in in addition to a full time job or full time combination of part-time jobs, when calculating child support.
Income for child support purposes does not include a spouse's income.
Child support obligations can be updated based upon a substantial change in these factors. So, unless the remarriage results in a change in income, in a child's extraordinary expenses, or in overnights spent, it isn't going to change child support.
Also, income includes income imputed to a party where someone could work but chooses not to do so. So, if a remarried mother quit her good paying job when she remarried, she might not necessarily get a reduction in child support since she could have kept her job and chose to drop it instead.
Obviously, this is an oversimplification, particularly with regard to imputed income. But, federal welfare statutes require states to have relatively similar child support laws for all but the most affluent people.
Alimony also known as maintenance or separate maintenance, historically terminated upon the remarriage of the person receiving it, or upon the death of either party to the former marriage. But, since this is not federally regulated (except with respect to its tax implications), it isn't as uniform among the states.
Remarriage of the person paying maintenance is not historically a ground to reduce or change maintenance obligations.
The majority rule is that the historic rule applies to remarriage in the case of court ordered maintenance that does not arise as a result of the agreement of the parties. But, the parties may, by agreement, provide that maintenance does not terminate upon remarriage.
This applies only to maintenance payments due in the future. One cannot, for example, obtain a refund of a lump sum alimony payment that has been made already because the spouse who received it promptly remarried even though the lump sum alimony payment was set based upon assumptions by the judge that the spouse receiving it would not remarry for a long time.
The minority and emerging rule is to allow courts to impose maintenance awards which are not terminated upon remarriage without an agreement of the parties to that effect. Another, basically opposite, minority rule is to forbid maintenance agreements where maintenance does not terminate upon remarriage. I don't have any good surveys at my disposal regarding which states follow which rule.
There is quite a bit of variation between states regarding circumstances where de facto remarriage involving co-habitation, etc. amounts to remarriage even though the maintenance recipient is not legally remarried. Some states will treat this as a de facto remarriage, others will consider this to be the kind of change in circumstances that allows a maintenance award to be modified, and still others will ignore de facto remarriage entirely.
In general, maintenance awards are much harder to modify than child support awards where modification is relatively routine when there is a change of circumstances. Usually, maintenance awards are made with consideration given to the income and broader economic circumstance of the parties when it is awarded, but is not a function of current income or expenses except to the extent that the original award because unconscionable or at least very unfair in the circumstances as they develop.
One factor that could make a maintenance award unconscionable and subject to modification is the inability of the spouse obligated to pay it to pay.
For example, support that the spouse obligated to pay maintenance was disabled and had a much reduced income a result - the maintenance award might be reduced.
The obligation associated with supporting a new spouse for the person paying maintenance would very rarely be considered a change in circumstances requiring a change in the maintenance award.
But, ultimately, this is a facts and circumstances test. I could imagine maintenance changing, if, for example, the new spouse was disabled and required extraordinary care that was costly in terms of both time and money, limiting the resources available to the person paying maintenance.