when I've been given a brief, I usually have a little brain storm session and come up with 3-5 different design ideas based around the key bits of the brief. Send these back to the client to make sure I'm going in the right direction. Then when they email back saying 'yep I like that bit of number 2 and most of number 5' I'll then combine said bits to make one logo, and then develop that logo idea further with 4 slightly different versions...
Don't be afraid to use google images to search for things like 'road racing' or 'racing logo' to see what other people have done and also to find yourself some inspiration. When you see a logo you like, take some time to work out why you reacted in a positive way to the logo, and if you wanted to create your own logo in that style, how would you make yours better?
Have a think about what 'road racing' encapsulates for you and your club members.. things like speed, engineering, teamwork, competition etc... can all be worked into the logo (design elements to suggest this stuff could include: clocks/timing numbers/abstract swoosh marks to suggest speed/silouettes of people working together/traffic cones/flags/bikes etc)
Tbh, the current logo looks very clean, concise and official. Is this the format of the club or is it more socially/commercially orientated?
Also as McK said, it's worth considering where your logo is going to be used. If it's just on websites & print then you can go nuts with a very intricate design and lots of colours or even graduated colours. If the logo is regularly stuck to bikes & cars in the form of adhesive vinyl then you might want to reconsider using so many colours or simplify the design slightly to reduce cutting costs @ the sign writers.
As for design software, I could only recommend the stuff I use (industry standard stuff) like CorelDraw x5, Adobe Photoshop & Illustrator, since I don't use anything else.
If you're struggling to come up with design ideas on your own, why not ask some of the committee to submit a selection of designs - this could save alot of 'paid' design time in a studio if you're all already agreed on the design and just want someone to create the digital version for you
