Starting a fiction magazine, what kind of contract should I use for submissions I choose to publish? Should I have a lawyer draft something custom?

Magazines generally use a simple form of agreement with the author. The author either (a) assigns the copyright in the work to the magazine, or (b) grants the magazine "first North American serial rights" (or similar rights as appropriate, which seems to be the model you contemplate). The model used depends to a great extent on the magazine's business model, how much negotiation "clout" it has with authors, and how likely it is the authors will be able to re-sell their works in a non-serial medium. The agreement will also usually provide for a flat fee for the author, as well as contain some warranties by the author regarding originality, no previous publication, non-infringement, etc. We represents authors and publishers in my office, and we'd be happy to discuss this subject with you. You should expect to spend about $1,000 or possibly a bit less on this kind of legal work.