Shapell's is a very popular yeshiva in Israel for (not exclusively, but mostly) ba'alei teshuva with limited prior learning experience. Their focus is getting people up to speed quickly, and many people are able to go from basically no knowledge to being able to keep up with those who have been learning gemara for their whole lives within a couple of years. Pretty much any level of knowledge is welcome as their tiered system works for people who can barely read Hebrew all the way up to solid gemara learners. The yeshiva is for post-college-aged men. There is also an affiliated seminary for women.
Kollel Nefesh HaChaim is an English-speaking Kollel in Jerusalem for young, married men who do not have extensive background knowledge in gemara.
Yeshivas Ohr Someyach is another relatively popular Yeshiva located in the Maalot Dafna area of Jerusalem which has several 'tracks' and programs of study for students of varying levels of proficiency. There is also an Ohr Someyach in Monsey, New York, which is also designed for beginners. It is a much smaller institution than the Jerusalem Yeshiva.
Aish Hatorah has some popular programs though i cannot say based on personal experience.