What is a good way to reason in physics? I have recently made the decision to study Physics seriously. However, in the past, I've had some difficulty with the subject because of my primarily mathematical background. I find that sometimes even if my reasoning is axiomatically correct, it doesn't lead to the right answer for simple reasons like I made a wrong assumption, or I hadn't assumed something I should have. So, I have three questions:
1) Is there some system or method I can use to determine what assumptions I should be making? This is important to me because it seems the problems I've encountered traditionally seem to be open-ended in this regard when they're not simple plug-and-chug questions.
2) There's also the question of definitions. I am uncomfortable with the fact that most definitions are not rigorous enough; this usually leaves some room for interpretation and consequently, it's unclear how they may be applied to certain situations. How do I know if the interpretation I've made of the definition is accurate?
3) Thirdly, when I'm reasoning with a problem, I'm not sure if at each step, what I've concluded is accurate or would actually happen in reality. This applies especially to problems where I'm asked to make simplifying assumptions (such as, "assume the surface is friction-less" or "assume the rod supporting the beam is mass-less") which simply wouldn't be applicable in the real world. So, I'm not really sure what to model my reasoning after in these cases; how can I get around this? Also, is solving such problems really helpful as far as learning actual Physics goes?
I can only speak for classical Physics, as I haven't progressed much further. Is Modern Physics the same in this regard, or is it more axiomatic? Finally, if there really isn't any systematic approach to this, then how can we really call the process Science? Isn't the scientific method all about finding agreement?
Thanks for all the help in advance!
 A: In my experience, the best way to go is to make enough simplifying assumptions that you can actually find an answer.  Once you've done that, then go back and reevaluate your earlier assumptions,a nd find a way to correct your model adding those effects back in.  
Alternately, figure out a regime in which your assumptions make sense.  In the case of friction, for example, friction is negligible so long as it doesn't remove energy for the system, so for a system with initial energy E, we can ignore friction so long as the distances we travel are less than $\frac{E}{F_{fr}}$, since, roughly (and not assuming a model of friction), The force of friction, if acting with a constant force $F_{fr}$ will drain out an amount of energy $F_{fr}x$ over a distance x.  good estimates make an assumption about what order of magnitude an effect will happen on, and then claim validity only in the regime in which that effect will have negligible impact.
A: 
1) Is there some system or method I can use to determine what assumptions I should be making?

I presume you are thinking of working in theoretical physics. Keep in mind that physics is not a QED discipline, it is a research one. For theory, if there is an assumption that is important it is the assumption of simplicity and economy in  modeling. That is why symmetries are so important in current theoretical physics. Next to it is acquiring a feeling for appropriate approximations that will lead to this simplicity in problem set up and solving, as is pointed out in another answer. This means you should be aware  in depth of the data that the problem is addressing.

2)  How do I know if the interpretation I've made of the definition is accurate? 

You will slowly acquire an intuition by trial and error . First by the grade you will get in the set of problems and then by solving real problems against data ( research). Sometimes you will be wrong, and will have to acknowledge it. That is what research means. 

3) Thirdly, when I'm reasoning with a problem, I'm not sure if at each step, what I've concluded is accurate or would actually happen in reality. 

It is trial and error, checking against data that will slowly give you confidence in the applicability of your approximations.

So, I'm not really sure what to model my reasoning after in these cases; how can I get around this? Also, is solving such problems really helpful as far as learning actual Physics goes?

You will not go wrong following  Jerry Schirmer's advice in his reply. You will slowly build up a data base that will allow you to make approximations within errors that describe reality/data.

Is Modern Physics the same in this regard, or is it more axiomatic? 

It is similar and more challenging because one has to build up new intuitive tools. 
The aim of the discipline of physics is to model the reality we observe and measure and use the model to predict new phenomena. There are laws that have to be obeyed, like conservation laws, but axioms are part of the hypothesis in finding a model for reality. If the data disagree, the axioms too will be changed.

Finally, if there really isn't any systematic approach to this, then how can we really call the process Science? Isn't the scientific method all about finding agreement?

Agreement with what? In my opinion the scientific method consists of constructing a hypothesis and using mathematical and logical reasoning testing it against data ; if it agrees, fine, if not, the hypothesis has to be changed.
