Given a circuit with resistors, op-amps, independent voltage and current sources, how do I solve the node voltages? Do I always assume that op-amps are ideal?
No, depending on what you are trying to do you might have to use a more complex op-amp model with finite gain, input offset voltage, input bias current, etc.
If it is a real problem you can apply experience and judgment to decide what can be left out. For set problems the rule usually is that you should use all the information given. If nobody mentions offset voltage, assume it is zero. There are exceptions- sometimes they'll throw in an irrelevant number (find DC voltages and they tell you the Gain-bandwidth product, for example), and sometimes you might be expected to parametrize the unknown variable.
There are 2 books/documents on opamps and electronics that would serve you well as a student, and will give you answers and approaches to your question:
Op Amps for Everyone, 2ed., Ron Mancini (Texas Instruments), 2003, Newnes, ISBN 9780750677011. This book is also available as a downloadable document SLOD006A from the Texas Instruments website - check for updates. The hardcopy book may be in its 3rd edition.
The Art of Electronics, 2ed., Paul Horowitz & Winfield Hill, 1989, Cambridge Uni Press. Available for webviewing on Paul Horowitz's website. Very good for circuit analysis and analogue electronics, a bit dated for microprocessors, but still a great book to learn from. Aka The Silver Bible.