There are just so many options in those neighborhoods, I feel like I could go on forever...

In Chelsea:

I LOVE Tia Pol and it is reasonably-priced for a tapas place, though if you're really searching for an inexpensive meal, tapas do have a way to add up...Especially these, which are so good!

In Union Square/Flatiron:

Just north of Union Square is Bar Stuzzichini, which offers a great deal. You can order 5 appetizers for $25 (for 2 people) or for 4-6 people for $49. I recommend the grilled octopus and the fried artichoke. The pasta is really fresh and great as well.

Also in Flatiron is Hill Country, which makes some of the best barbecue in the city. I've been several times and always have a blast.  There is great live music and its never costed me much since I share a lot with friends.  They usually have a special combo deal for two people but I can almost guarantee it will be enough for 4. Save room for the bourbon pecan pie and DEFINITELY get the moist brisket.

In The West Village:

'ino serves really reasonable Italian panini and bruschetta. It can be hard to get a seat but the wines are great and the food is top quality.

Wild Ginger also has extraordinarily cheap Thai food.  Based on the decor, it seems like it would be more expensive, but somehow it isn't. 

I hope these help!