you can consider transistor nothing more than a device which help you in controlling the parameters or, say, circuit 2 with the help of circuit 1(just a rough estimation) if transistor is joining the two circuit. For eg. as in digital electronics there is a clock pulse and say you want to do something when the clock is at a particular level, similar is the case with transistor, you can model the transistor so that at the operating point when the voltage at the base reaches a particular level then you can turn the device on and so the current can flow in the ckt2, or you can think of it as a relay, or a switch, not only this transistor is an amplifier.
for designing purpose just keep in mind that transistor helps you in controlling the parameters of the circuit 2 with the help of ckt 1, so for determining the operating point you can use any model. Don't get confused with the different models available for solving transistor these models are just for you convinience, it is easier to use the re model as it facilitates easy computation, h-parameter(hybrid) model is the most versatile and is considered the best in solving any transistor, but T-model is also good.
to get a basic feel of what a circuit is doing you can approximate using the approximation like the Vbe = 0.7 and all these all approximations lead to easy computation.
i know two very good books on studying transistor
1) electronic devices and circuits, boylestad , a very good book, but it uses a lot of approximation and is good for somewhat approximate analysis
but if you want to model the transistor in detail like you want to know the exact parameters and all then there is a better book
2) microelectronics circuits, sedra smith. this you can call a bible, super book, but i would advice to first read the book 1, then move on to 2, otherwise you won't be able to learn much and you will just bury yourself in complex mathematics.
for learning how to solve how to analyse circuit study as many circuits as possible and then with the passage of time you will come to know how can you use transistor in many different ways
for learning this you can refer to books written by forest m. mims
they contain just circuits. and you can analyse them.
FET is not much different from BJT, its just FET are mainly used for making of amplifier because of its very high input impedence but the output impedence is almost comparable, it is also small in size, but on the contrary BJT has high switching power so if your application has to do something with switching BJT would be a great choice.
at last i would again say, if you want to learn transistor then study a lot of circuit may be you can look into the construction of op-amp as they are nothing but 4 stage differential amplifier and through that you can also learn..
have a nice time learning TRANSISTOR !!!