Several of the grammars of Old Norse are actually readers, primarily, with a grammatical appendix. Below are a few useful resources. The bibliographies of these contain references to more scholarly text editions.
An Introduction to Old Norse, by E.V. Gordon; Oxford University Press, Oxford, 1990 (repr.). This is still regarded as the standard reference, though often extremely daunting for self study. Texts are first, with grammar tucked somewhere at the end. The text selection is nevertheless fairly broad and engaging.
Edda. Die Lieder des Codex Regius nebst verwandten Denkmalern, ed. by Gustav Neckel and Hans Kuhn; Carl Winters Universitaetsverlag, 1968. Actually two volumes, the first containing the text of the Codex Regius, the second a complete glossary. The standard edition of the Poetic Edda. The glossary has been translated into English (see below).
Below are a few grammars worthy of note. Most are out of print, and few are in English.
Altisländische und altnorwegische Grammatik, by Adolf Noreen; Max Niemeyer Verlag, Halle, 1923. Still the standard grammatical reference for Old Norse grammar. Purely a reference grammar.
Altisländisches Elementarbuch, by Andreas Heusler; Carl Winters Universitätsbuchhandlung, Heidelberg, 1932. A concise grammar in logical form, not graded lessons. Places a large emphasis on historical sound changes, e.g. quoting verb paradigms with -er instead of the more common -ir, and nominal paradigms with -om instead of -um. Though not the current pedagogical trend, this is in fact quite useful as preparation for using Neckel & Kuhn's Edda, which retains manuscript readings. This grammar has short text selections and a glossary in the back. Its greatest feature, however, is a very detailed account of Old Norse syntax. The book is worth consulting for this feature alone.
Grammatik des Altisländischen: Mit Lesestücken und Glossar, by A. Zaluska-Stroemberg; Helmut Buske Verlag, Hamburg, 1982. A wonderfully written grammar, though in logical form rather than graded lessons. Very nice for the constant attention to matters of etymology. Contains a very broad spectrum of short text selections at the back, the first with translations (in German), the later ones without. Includes a glossary (ON to German).
Old Icelandic: an Introductory Course, by Sigrid Valfells and James Cathey; Oxford University Press, Oxford, 1982. The only grammar of Old Norse (restricted to Old Icelandic proper, though this is not much of a restriction) set out in a series of graded lessons. Extremely user-friendly, with a wealth of grammatical information. Now out-of-print.
A New Introduction to Old Norse, by Michael Barnes; Viking Society for Northern Research, 1999. This book, together with the Reader and Glossary by Anthony Faulkes (in separate volumes), serves as an excellent introduction to the language for the novice. Though not laid out in lessons like Valfells and Cathey, Barnes' book nevertheless proceeds gently to lead the beginner through the novelties of languages with a robust case-system and verbal morphology. A nice introduction for those with limited foreign-language experience.
Two of the most useful dictionaries for those just beginning their study of Old Norse are the following:
A Concise Dictionary of Old Icelandic, by Geir T. Zoega; Oxford University Press, Oxford, 1972. A smallish dictionary, but with ample vocabulary for all but the most obscure texts. Particularly useful are the charts in the back containing morphological paradigms and lists of irregular verbal and nominal forms. As with many of the most useful books in Old Norse studies, but most frustrating in the case of this one, the book is more often out of print than in.
Glossary to the Poetic Edda, by Beatrice La Farge and John Tucker; Carl Winter Universitätsverlag, Heidelberg, 1992. An incredibly useful translation of Kuhn's glossary. A wonderful gift to the study of the Poetic Edda.
The following books study the relation of Scandinavian languages to the broader group of Germanic languages:
The Indo-European Languages: An Introduction to Their History, by Einar Haugen; Harvard University Press, Cambridge, 1976. An authoritative study which has become a standard reference in the field.
Old English and its closest relatives: a survey of the earliest Germanic languages, by Orrin W. Robinson; Stanford University Press, 1992. An impressive book for the vast amount of material it covers in such a small space. Conversational in style and very engaging. This book is ideal for students wishing to get a jump-start into the field of Germanic historical linguistics.
A wonderfully readable account of Scandinavian customs and practices during the Viking Age may be found in the following:
Everyday Life in the Viking Age, by Jacqueline Simpson; Dorset Press, New York, 1967. A very readable outline of Viking Age society and culture.
There are several works on the language of the runes. The most recent standard on the language of the earliest inscriptions, and the obvious source for the lion's share of runic material in these lessons, is the following:
A Concise Grammar of the Older Runic Inscriptions, by Elmer H. Antonsen; Max Niemeyer Verlag, Tübingen, 1975. An absolutely exceptional work, laying out clearly the system of writing as well as the morphology and derivation of the language.