You may find the jump from Spades to Contract Bridge much larger than you expect. However the popularity of (generic) Bridge pre-dates the invention of Contract Bridge by Harold Stirling Vanderbilt in 1925. If you have a gaming group interested in advancing their card-play skills together, you might consider jumping first to one of the antecedent games from the turn of the Twentieth century, the games that initially inspired early stars such as Sam Stayman and Oswald Jacoby. In chronological order (and thus also in order of increasing complexity):
as well as a more recent related game:
Although the Laws of Contract Bridge are straightforward, at least until you have to deal with irregularities, that is not the source of the game's complexity. Many great minds have worked over the past century developing increasingly complex bidding systems and conventions, and one must know at least one of these in substantial detail before setting foot in Bridge Club or Tournament with intent to play.
However, when your group decides to jump into Contract Bridge, you could do substantially worse than to use an old reference by either Ely Culbertson or Charles Goren. Both of them were extremely accomplished players from the early days of Contract Bridge, and wrote extensively to popularize the game. Both advocated a natural bidding style with similar features (which came to be called "approach-forcing"), the main difference being the use of the more accurate Work Point Count by Goren to replace Culbertson's Quick Trick hand assessment. Old copies of their books are still available at used book stores now and then at very reasonable prices.
Many national and other bridge organizations also publish guides on How to Play Bridge:
Comments being ephemeral and all that, the astute comment below by Mycroft is here reproduced:
To emphasise a point mentioned, but not really pointed out, in the answer - DO NOT try to learn Bridge by reading the Laws. Unlike many board and card games, they don't even give an idea of how to play - as stated, the language(s) of bidding are necessary to even start play, and nothing in the Laws explains what things mean - just who can decide what they're allowed to mean. In this respect, it would be more like learning to play hockey by reading the NHL rulebook. In both cases, it's a tool for referees rather than an introduction to the game - very useful for veterans, but not newbies.