Well this post started out as a collection of ideas for your math club but turned out to be an outline for the entire club so here it is.
Have as many incentives to participate as you can think of such as having a point system where you earn points for doing better on competitions. You could see if math teachers would give extra credit for doing math team but at the very least get the support of all the math teachers to advertise the club. To recruit students you could pose a problem of the month/week or post teaser problems in your school newspaper that will be answered in upcoming lectures.
Definitely hold competitions for the club on a regular basis. Most people on the team won't want to practice regularly if it isn't preparing for an up coming competition. Experiment with the format of the competition, but some commonly used formats for competitions include [for individual competitors:] "countdown" where you have to answer as fast as possible, and just a regular timed test where the people with more correct score higher and [for teams] "GUTS" where a team is given one or more problems at a time, they pass the problem in when they're done and receive a new problem but can't go back to old problems and "relay" where each person answers questions on their own and the answer to one problem is passed off as the value of as a number needed for the next problem (for example, the first problem is to solve for x in 4x+2=4 and the next problem is to compute the length of a diagonal in a square with a side length of ? where the ? is the value of x in the previous problem). You won't have to write the questions yourself, you can get them on the internet and in problem books.
Participate in local competitions as well even if they are relatively low quality just because it's a fun way to get the team together for a "real deal" competition. Competitions like ARML, PUMAC and HMMT will probably be out of reach for most members of the team at first but you might want to eventually send a team to these competitions because traveling is fun (you can carpool). Do the contests that can be done from school like AMC, mandelbrot and maybe mu alpha theta. Another good contest is USAMTS which is an online proof-based contest.
You might consider splitting up the team into two tiers -- one for younger students and one for older students -- in order to ensure that the material isn't too hard or easy for anyone (don't actually have two different clubs unless your school is large enough to support that). You could hold a competition for middle schoolers nearby. This way you can have the entire club participate in the writing of questions (they can't write the questions for a competition that they participate in!) which is a great way to understand how the problems on competitions should be approached especially for freshmen whose problem solving ability might be just above that of most middle schoolers.
Use meeting time effectively. You could have optional meetings during lunch to collaborate on solving problems but don't spend too much time solving problems at the main meetings because students can do that at home and will probably want more out of the meeting than collaboration, such as presentations and mini-competitions. You might go over the solution to a posed problem and have some warm-up problems but don't use the whole time for collaborating. Don't make the meetings seem too much like a math class because that will turn off most people (even if the material they are learning is challenging and fun). The meetings should overlap as few other extracurricular activities as possible and ideally should be held directly after school so kids don't have to be motivated to drive all the way to school to participate. Above all have food at the meetings.
You could set up a website to post problem sets to, but try to involve the students by, for example, offering points for correct solutions otherwise few will do the problems. You might also consider allowing students to post problems to challenge other students; wordpress with \LaTeX works well and solutions could be posted as comments. The reason that I don't say to use a forum is because without a lot of experienced members to regulate content, it might turn into a collection of nearly-impossible problems that kids post just to come up with the hardest problem. To prevent this require the students to submit to submit a solution along with the problem.
I wouldn't suggest trying to do the things in this post at once when you're starting the club because it would too much. Start out slow and see how people respond to new ideas because most of the things you do at a math club depends entirely on the group of kids who joins the club.
