Is starting a career at a major pension fund a good place to learn about investing.

The names you mentioned are basically equivalent, in many ways, to starting at an Asset Management firm. They have their own internal staffs, investment teams, etc. and more often than not will have an investment consultant over top of all of it whose tasks will vary. 

Most of the junior type roles are a real mix of tasks - that could be interfacing with the consultants they work with on the plan, supporting things like attribution/performance reporting, manager analysis, economic research, or depending on the strategy/asset class supporting the direct investment of assets. For the most part - what I've seen is you'll get a broader experience, more as an asset allocator than a direct investor - and as you go, you can specialize or take advantage of opportunities that exist. Your mileage will vary - it very much depends on the fund, team, and even the state - at that level, there are politics involved that usually don't roll down too far but it can impact overall stability and trajectory. 

Generally - the exits are a few places. Consultants are one route, where you take your experience to one of those (i.e. Callan, Mercer, Cambridge, etc.) or an OCIO provider (i.e. discretionary consultant). Most likely you'll have experience on the more macro side, manager research, due diligence of asset classes, setting up a strategic asset allocation, making changes to the portfolio, sourcing/implementing investments. If you end up in a role where you are working on directly investing - say that FI example I used earlier - you might have additional opportunities. You'll also build out a pretty good network - people move around and it can be a good place to get experience + network to leverage. Certainly, if you are only there a few years, you can get a CFA and then take that experience to other roles, firms, etc.