In Israel the most common way of converting is via the Rabbanut, which imposes a strict 1 year requirement: you must spend at least 1 year going to courses organized by them, supervised by them.
Machon Meir does a special conversion track, I believe.
*However* in your situation, seeing that you are already an Israeli citizen, you *can* also undergo giur through a private orthodox beis din. This is more relevant if you want to be chareidi, really. (From the way you describe it I’m guessing you’re more like Modern Orthodox).
There are several chareidi botei din in EY that deal with giur, without necessarily being recognized by the state. The most prominent is the beis din of Rav Nissim Karelitz in Bnei Brak. If you undergo giur there, you *can* subsequently get registered as Jewish by the Rabbanut and get married via the Rabbanut. (However, you cannot get apply for Israeli citizenship/residency – ‘aliyah’ based on that; but that doesn’t apply to you since you’re a citizen already.)