I"Lel go with Smalls also, they have an open mic after the band plays. My son sits in on drums! Also the place next door Mezzeros or something like that.
Everyone likes Small's but it IS really small and the one time I was there the seating was on benches and it was really cramped. Nobody has mentioned the Village Vanguard which is my fave, no food, just drinks and great music. Uptown Smoke Jazz and Supper Club is great, near Columbia U. Blue Note has greatshows, decent food, drinks, top dollar but great shows. Dizzy's (Jazz at Lincoln Center) also has great shows, good food and drinks, great view of the park. In NYC you can see potentially three shows in one night, easily two. One night I caught the 8 oçlock at Dizzys, walked up Central Park West, did a number or two, caught a cab the rest of the way to Smoke for the midnight show. Lots of combinations like that are possible. Of course, it's NYC and it costs $$.
For great live standup comedy I like "The Comedy Cellar" near NYU. at least 3 shows a night but you have to call ahead for a reservation and pay a small entry when you get there.You'll see 5 or 6 top name comics per show. It's 2 doors away from the "Cafe Wha" where Hendrix used to play before he was "discovered" by Chas Chandler. The oldest (and best) pizza place in New York is also not too far from there. unfortunately i can't recall the name of it. But it was truly amazing.
Fat Cats is a fun place to hang out, low cover charge ($5) and an eclectic mix of jazz, blues and gospel. Big place but much of it is taken up by billiards, table tennis, even backgammon tables. Bands (often made up of a mix of seeing veterans) play in a cozy place behind the bar, where you can sit on a couch or living room-style furniture. Most of the musicians get paid in a 'pass the hat' kind of mode. The Gospel Queens are amazing on weekends.
Brooklyn Bowl (this is the original one) is great. Bowling alley and concert space. Bands often take up a residency while in town recording in Winter.
Or stay in lower Manhattan and catch the original City Winery. Generally eclectic mix of acts....Ronnie Spektor to Ian Hunter to Peter Wolf to Stephen Stills. NOT cheap. Tix often $45 or more. Or wander in to the restaurant side of the club, sit at the bar, listen to the show and view it on-screen at the bar for free. If it isn't sold out, be nice to the bouncer and they may let u slide in.