Well sure if you can't do dns then internet doesn't work very well ;)
Do a simple query from a client with your fav dns tool, nslookup, dig, host, etc.
Does www.google.com resolve? Can pfsense resolve - go to diag menu on pfsense dns lookup for say same www.google.com do you get back an IP?
Where are you pointing clients for dns? Pfsense IP I would assume? Out of the box pfsense would resolve for dns, vs forwarding..
Not sure what your trying to accomplish with those switches that seem to be doing routing? Where do the clients point for gateway? The switch IP or pfsense? What IP do you have on pfsense in those vlans? Sure hope its not .0?? How exactly are you routing between the switches - you show them connected with their e0/1 interfaces - but you list no IPs on them - is that a transit network?
On a side note - why the use of /20?? Such a mask makes no sense in a lab setup, why would you not just use /24s? Makes it much easier to tell where the network breaks, etc.