There are two ways ot formally propose amendments:
1. 2/3 of both houses od Congress agree on a porposed amendment
or
2. 2/3 of state legislatures ask Congress to call a national convention to propose amendments. (This method has never been used.)
Once an amendment has been proposed, it must be ratified. Either:
1. Three-fourths of the state legislatures approve it
or
2. Ratifying conventions in three-fourths of the states approve it. This method has been used only once -- to ratify the 21st Amendment which repealed Prohibition. (Prohibition was itself a Constitutional amendment.)
The Supreme Court has stated that ratification must be within "some reasonable time after the proposal." Beginning with the 18th amendment, it has been customary for Congress to set a definite period for ratification. In the case of the 18th, 20th, 21st, and 22nd amendments, the period set was 7 years, but there has been no determination as to just how long a "reasonable time" might extend.
The president cannot veto an amendment or negate ratification.
There tons of ideas for amendments. Hundreds surface in Congress each year.
Of the thousands of proposals that have been made to amend the Constitution, only 33 obtained the necessary two-thirds vote in Congress. Of those 33, only 27 amendments (including the Bill of Rights) have been ratified.
The 6 that were never ratified are listed here on Wikipedia.
Some proposed Amendments lists all sorts that have been considered by congress.
I'm not aware of some that are seriously being considered right now, although the Equal Rights Amendment will probably come back. There has also been serious talk about changing the provision that to be eligible for the presidency, you must be a "natural born citizen."