Tri-Start simply removes chlorines and chloramines. It has no bearing on whether or not a tank is cycled. As far as I know, BIO-Spira is the only product proven time and again to reliably and safely cycle a tank within a week's time.
Most folks on the boards here will recommend a fishless cycle. Most people are too impatient their first time through, though. In addition to that, many folks who fishy cycle once vow never to do so again. Let me explain a few key benefits to fishless cycling. First, you don't harm or kill fish. Yes, in a fishy cycle you can do everything right and luck out sometimes and not kill a fish. With a fishless cycle I can guarantee you you will not kill a fish during your cycle if you do it correctly which is to say, not adding fish until your tank cycles.
Second, with a fishless cycle there are no daily 50% water changes. Trust me, chnaging 5-6 gallons of water every day for 2 weeks may not seem like much, but once you're on day 7 or 8 you start wondering why on earth you didn't listen to those fishless cycle geeks. With a fishless cycle you simply test water, add ammonia as needed, then kick back with a refreshing beverage while you surf the internet looking for the right fish. During fishy cycles you test the water daily, maybe do a water change, test the water again, maybe do a second water change because the stupid ammonia levels are still too high and your fish are breathing rapidly... by the time you're done you don't want a refreshing beverage because it reminds you of water!
Third after a fishy cycle you can only safely add fish one or two at a time every two weeks or so and still have to carefully monitor your water with tests daily for months while you build up to your normal stocking level. With fishless you can add all of your fish at once as soon as the tank is cycled and you've made the one, count it one, water change. After that, daily testing for a couple of weeks to make sure you're steady, then you can taper off.
In the long run fishless is better because it is cheaper (you use less testing supplies as well as water and time), easier, takes less of your valuable time, allows you more freedom of stocking than does a fishy cycle and the kicker, no fish are harmed.