My professor advised us to get either Hardy&Wright or Ireland&Rosen for our introductory number theory course. I would like to ask what are the differences between these textbooks in terms of pedagogical effectiveness; completeness; relevance; difficulty; material covered (also, some personal thoughts and advice would be really good). Are there other books which I should prefer to the ones mentioned above?
Some caveats: I own both books, and have taken number theory courses up to graduate level. Also, your questions are clearly somewhat subjective: what is difficult, relevant or complete for one person, depends on their previous exposure, inclination, course content, and ability.
So...in my opinion.. both books are classics, which means they have been around for some time, have been reprinted, corrected and reprinted again, and used by many students all over the world. HW is certainly an introductory text: you could give this to an able 16/18 year old for them to work through, from chapter 1. The treatment is comprehensive and is logically ordered. I found the notation comfortable and it more than covered my first course in number theory.
The IR book covers slightly more ground in fewer pages. I think that last statement accurately sums up this book. My appreciation for this book only grew once I had covered a number of other courses (on groups, rings and fields) - and matured somewhat in my mathematical thinking (for example there is excellent treatment of quadratic reciprocity). Indeed, IR's book is an excellent taster of algebraic number theory - but that is unlikely to feature in an introductory course.
In short, both books are excellent. I own both. I mostly used HW for my introductory course. As a bridge to more advanced levels, IR's text is excellent if you already know you are interested in algebraic number theory.
Ireland & Rosen is a graduate textbook which starts with general number theory and moves to algebraic number theory. Abstract Algebra is not strictly required, but students would benefit from a course in it. The first half of the textbook can be used, with some care, as an advanced undergraduate textbook.
Hardy & Wright is an introductory number theory text. It's suitable for undergraduates or perhaps advanced high school seniors (first few chapters). The latter half of the book could be adapted for use in a graduate class, especially in combination with another book.
Very subjective: Ireland & Rosen is more modern and better-organized. Hardy & Wright is more eclectic and perhaps more fun. Both are good textbooks, but I don't have enough experience to compare them broadly to others.