Hi, it all depends on what you want to pay really. The lower priced ones will be hard to find on the net. If you're looking for 10k baht or under I'd rent a place for 2 weeks to get you started, then look/ask around once there. Searching "phuket apartment long term" on google may help.
Edit: I can't post links, remove the space between www and the "." to get the right address.
Search by apt or house for rent, city Phuket here:
classifieds.thaivisa.com/advanced_search/
www .phuketgazette.net/classifieds/showbycat.asp?Kat_Id=34
phuket-post.com/classifieds/index.php?catid=1&set_add_ad_cat=
www .thailand-classified.com/index.php?ct=rlestr4& (check phuket)
www .siamrealestate.com/search/?search=1&as_province_id=53&as_otype=rental_longterm&as_ptype=0&as_bed_min=0&as_sale_structure_price_min=0&as_sale_structure_price_max=0&as_sale_land_price_per_rai_min=0&as_sale_land_price_per_rai_max=0&as_rental_holiday_price_min=0&as_rental_holiday_price_max=0&asrh_di=&asrh_do=&as_rental_longterm_price_min=0&as_rental_longterm_price_max=0&as_text_matching_text=&as_text_matching_fld=p.code&as_text_matching_rel=contains&x=32&y=16
Western style places:
www .chalongapartment.com
phuketdir.com/thepierphuket/
Good luck, and post back if you find anything interesting, I'll be over for another year come this summer.