Hopefully, your choice of whether to marry a U.S. lawful permanent resident (a green card holder) or a U.S. citizen will be personal, not based on immigration considerations. You will be eligible for an immigrant visa and green card in either case. Nevertheless, marrying a U.S. citizen is definitely the faster path to a green card, as described below. And if your U.S. spouse is currently a permanent resident, this will help you consider whether their applying for citizenship as soon as possible makes sense.
If you are married to a U.S. citizen, you will enjoy many benefits, but also face a few inconveniences.
Spouses of U.S. citizens are eligible for an immigrant visa (if needed for U.S. entry) and then a green card (lawful permanent residence) under the Immediate Relative category. This category has no numerical limit on the number of visas issued each year, which is important because you can't possibly find yourself facing a situation where visas have run out for the year and you must wait until one becomes available. In the past, the wait has ranged from zero time at all to an average of two years.
Getting through the green card application process is nevertheless likely to take several months at least, starting with filing an I-130 petition. There is no getting around that aspect of U.S. government bureaucracy, regardless of which category of green card you apply within.
A non-citizen who entered the U.S. legally, most likely on some sort of temporary visa (and not with the intention of doing so in order to apply for a green card) has a unique advantage. That person is likely to qualify to use a procedure called "adjustment of status," in which all paperwork (the I-130 included) can be filed at the same time with U.S. Citizenship and Immigration Services (USCIS). The agency will process it all and schedule the immigrant's green card interview within the United States.
That can be particularly helpful for someone whose permitted stay under a visa has long since ended. As described in Consequences of Overstaying on a Temporary U.S. Visa, this could lead to an inadmissibility problem if the applicant were to leave the U.S. in order to apply via a U.S. consulate. Being able to adjust status fixes this issue, because there would be no need to depart the U.S. before getting the green card.
After you are approved for permanent residence based on marriage to a U.S. citizen, the speed with which you got through the process could introduce one inconvenience: If your marriage was less than two years old at the time you became a U.S. resident, your residence will initially be only "conditional," and you will see a two-year expiration date on your green card.
The idea is to give the immigration authorities a second chance to judge whether your marriage is bona fide, or the real thing, not just a sham to get a green card. For more information on this part of the process, see How to Manage Your Conditional Resident Status.
There's yet another benefit to being married to a U.S. citizen: Three years from the date you become a permanent resident, you can apply for U.S. citizenship, so long as you remain married to and living with the citizen all the way up to the swearing-in ceremony. Most green card holders have to wait five years before applying for U.S. citizenship.
If you marry a U.S. permanent resident, you must apply for an immigrant visa under the Family Second Preference category (F2A). Your experience will be different than that of someone applying based on marriage to a U.S. citizen.
The number of visas/green cards issued each year under category F2A is limited, and the demand is often higher than the supply. In fact, it's rare for applicants to receive a visa number (as is required in order to get a green card) within the year their U.S. spouse first files the I-130 visa petition for them. Instead, they are typically put on a waiting list, based on their "priority date" (the date USCIS received their spouse's I-130 petition).
The waits are often around five years long, depending in part on which country the applicant is from. (Per-country limits apply, and demand tends to be particularly high from India, Mexico, China, and the Philippines.)
You must wait for your priority date to become current before you can take the next step and apply for your immigrant visa and green card under this category. (Learn to track the progress of your priority date.) Then, like people who are married to U.S. citizens, you will probably have to get through some months' more worth of application processing.
Unlike spouses of U.S. citizens, immigrants married to permanent residents, even if they entered the U.S. legally, will not qualify after an overstay to use the "adjustment of status" procedure and file all paperwork with USCIS, never having to deal with a U.S. consulate at all.
With rare exceptions, the immigrating spouse will need to leave the U.S. for a visa interview at a U.S. consulate (called "consular processing"); and possibly face three or ten years of inadmissibility barring return, as described in Consequences of Overstaying on a Temporary U.S. Visa.
One benefit of F2A applicants sometimes spending years on a waiting list is that, if two years have passed since the wedding when they're finally approved for U.S. residence, it's likely to be permanent, not conditional residence. They won't need to go back to USCIS in two years with another application proving the marriage is ongoing in order to become permanent residents.
Given all these factors, it would probably be beneficial for the permanent resident spouse of a would-be immigrant to look into applying for U.S. citizenship as soon as eligible. More information on this can be found in Steps to Become an American Citizen.
If you're looking to get a family-based green card, you might want to talk to an attorney to find out if they can help with the process. Hiring an attorney to help with the petition process adds legal fees, but a professional may help to expedite your case and avoid any legal problems.