The stock Esprit uses a non-intercooled Garrett T3 Turbocharger with an external wastegate set to 8psi on carbureted engines and 9.5 psi on HCI cars.
Overview of how turbochargers work
The Garrett T3 is a popular turbocharger and is still used in many aftermarket setups today. The exhaust housing may have to be transplanted from your old turbo to whatever replacement/upgrade you get. The stock Turbo is a Garrett T3 with a 40 trim impeller and an A/R 0.48 exhaust housing.
This means that most parts are inexpensive and easy to find in many places. A new turbo can be found for under $300US.
If you need to have your turbocharger rebuilt, Majestic Turbo in Waco, TX does a fine job for $335 (which includes return shipping).
Increasing boost means keeping the wastegate from closing when it "reads" 8psi (on a carbureted engine.) There are three methods to do this:
Increase spring strength in wastegate
Increasing the spring strength in a wastegate can be done by replacing the spring (not sure if you can do this in the stock wastegate,) or replacing the wastegate with a wastegate with a stronger spring. This can be the simplest way to increase your boost (since you just have to unbolt the existing wastegate and bolt on a new one.)
Bleed off a some of the boost manually
If you bleed off some of the boost that is used to trigger the wastegate you can trick the wastegate into "seeing" less boost than the car is actually making. The best way to do this is to install a manual boost controller. Manual boost controllers are very inexpensive, but tuning them properly can tend to be problematic.
Install an electronic boost controller
A boost controller requires that you install a electric solenoid in the boost tube running from the turbo to the wastegate. Using the solenoid, the boost controller can "hide" the boost level from the wastegate until the controller wants to open the wastegate. This is actually a method used by many modern turbo-charged cars.
The simplest of boost controllers allow you to set a boost level and forget about it. There are boost controllers with fuzzy logic and programable aspects that allow you to choose various programs, increase (or decrease) your boost level at different RPM levels (so you could run 12psi at 3kRPM and 9psi at 7kRPM.)