The major phone companies in Taiwan are Chung-Hwa Telecom (中華電信), Taiwan Mobile (台灣大哥大) and Far East Tone. You can find offices all over Taipei, but the closest to the institute are in Gongguan. Mobile phones in Taiwan are unlocked (i.e, you can use any SIM card on them) and you can buy them either from any of the companies above or from many phone shops (e.g., GuangHua Digital Plaza 光華商場 or NOVA 資訊廣場 shop). To get a phone or a SIM card you need to go to one of the branch offices of one of the phone companies. You can either use a pre-paid option or a contract.
Most foreigners use prepaid (non-contract) phone card / 3G services. You can get a SIM card from one of the companies and then buy phone or 3G credit at any 7–Eleven in the Ibon machine, or top up online.
If you want to go for the contract, you can get a phone for a significantly cheaper price. You are normally asked to stay for 2 years within the contract. For a contract, holding an ARC is needed and normally either a local Taiwanese has to assist you (just to prevent you running away with the phone) or you can pay some extra fee (a couple of thousand NTD) that you can in principle recover after the two years period.