I'd hook up with some of the guys racing in NASA or SCCA and see what the safety specs call for in terms of size/wall thickness. It will be chromoly tubing (higher yield/tensile strength) for more strength and less weight. Also you refer to 'cage' in the 'back'. Sounds like you're talking about a roll-bar more than a cage. That will help stiff things up, but not as much as a full cage that ties in 6 or 8 body points to the structure. The challenge is always that a really stiff/strong cage is properly triangulated - and when you do that, you pretty much rule out use of the back seat in any reasonable manner. You should have sub-frame connectors welded in, and the mounting points of the cage should be reinforced with steel plate, and they should tie in to the floor and surrounding structure in 2 dimensions. That is, the reinforcement plate should have significant weld area in both the horizontal floor pan, and the vertical rocker panel section. Same is true for any tie-in points - in the foot wells, firewall, or trunk - tie into two dimensions or planes for the most rigidity.
Be sure whoever you have helping you knows how to weld high strength steel (it's a bit trickier than mild steel), and understands a bit about structural design and uni-bodies. Most importantly for a street car - be sure that any steel structure that is within a 12-18" of a part of you or your passenger's body is padded with proper roll cage padding. Can't tell you have many injuries occur because a body part - head, hand, leg - gets slammed into an unprotected steel surface. Given the forces that can be generated, bones and even a skull can be fractured in a very low speed accident. Soft tissue damage can occur even during spirited cornering - taking a buddy along for an autocross pass. I still have the mark on my helmet from where it slammed into the upper cage bar on a particularly violent turn-in in a special 2-seater Sports2000 SCCA racer in which a professional driver was treating me to a 10/10ths lapping session as a passenger at a driving school. People think they're making the car safer, and they end up hurting themselves worse because they've placed an unprotected non-forgiving surface (roll cage component) right where a soft body part can be slammed into it. Body part loses that battle every time.
