The subject of how Spice handles all the different analysis requests would fill a book. In fact, it fills books! There is no possible way to cover all of that here. (Nor can I pretend that I completely understand all of practical details involved.) However, I can consider the idea of addressing transient analysis, since I do understand that part of Spice better.
First off, I'd recommend that you purchase a copy of Laurence W. Nagel's (who is, to all intents and purposes the "father" of Spice) "Spice2: A Computer Program to Simulate Semiconductor Circuits," Memorandum No. UCB/ERL M520, 9 May 1975, which is available from "Electronics Research Laboratory, College of Engineering, University of California, Berkeley, CA 94720. (The title is slightly misspelled on the document you'll get, saying "Semicoductor" instead of "Semiconductor", but oh well.) This is the seminal and definitive document on Spice's operations. If you want the details, it's all there.
But the overview for transient analysis is something like this: Ideal branch relations are developed, constrained by explicit and implied Kirchoff laws. Then, cutset and loop analysis is applied. (Which, by the way, place limitations on the types of branch relations that can even be included in the circuit. So there are some kinds of equivalent replacements that take place about this time.) For formulating, a modified form of Nodal Analysis, or hybrid analysis (state variable analysis, for example), or the sparse tableau method (all variables in a huge vector and a very, very sparse system of circuit equations) are used.
In the end, numerical methods are also required of course. Some of the details also involve different numerical methods to solve, once all the formulation is complete. Many Spice programs include multiple approaches to numerical solutions. There are various problems that arise here, as the problem is basically very much like an infinite plane with all kinds of local minima and maxima and a very, very difficult time locating and finding the global maximum or minimum across that entire plane. Spice may get "trapped" into a local high/low point and not be able to find its way out of there. So there are other techniques which can help either recognize these issues and apply "tests" to see if it needs to widen the scope (source-stepping methods) or else start up multiple solutions and see which of them find the right global result (simulated annealing.)
A lot has happened over time and not all Spice simulators use all of the exact same tools and/or all in the same way, all of the time. This is one of the reasons that professionals may include several different implementations of Spice in their toolchest or, in the case of IC manufacturers, will have programmers on staff who continue to develop and modify Spice so that it handles cases they care about well.
But seriously, if you are really interested in understanding exact details (including examples provided), then I'd recommend that memorandum I mentioned above. It's not terribly expensive and it is, far and away, the best source for explicit details about how Spice works (with references for still more details, if needed.)