China TESOL Network are recruiters...and thus should be approached with caution if at all. Recruiters have a ROTTEN track record in China. Please consider not using one. You don't really need them anyway.
The school deal doesn't seem too bad on the surface, but...
- it MUST include a residence permit
- the apartment should be fully private, not shared...and don't be taken in by the words "private bedroom"!
- the acceptability of the hours depends a lot on the number of hours- classroom and otherwise- you're required to spend lurking about the school.
- I REALLY HATE these schemes where you work cheaper for a few months before you get your full salary. They are IMHO out-and-out stealing...you still have to do the same work, and they don't refund you the difference in salary if they decide to retain you after the "trial period". Schools do this purely to cut their costs...and for no other reason. Negotiate this one out or keep looking.
Taizhou is on the coast of the southern half of Zhejiang Province (which borders Suzhou's Jiangsu Province). It's fairly big...probably in the same size range as Suzhou. It's not a terrible place, but it's a pretty new city and doesn't have a lot of local architectural or historical color. It's far enough away from Hangzhou, Ningbo, and Shanghai to fall back into the "rather provincial and boring" category. Environment is so-so...being near the sea helps, but then one of their major local points of pride is their immense bank of coal-fired power plants. There's a fair amount of heavy industry there. However, it's also something of an agricultural center, and there are a number natural-scenery-type spots in the region. Zhejiang is very rich, and the economy in Taizhou seems to be sufficiently robust.
Upshot: it's not even close to an A-List city...but you could definitely do a lot worse.
However, it isn't within easy striking distance of Suzhou. Currently all trains from Zhejiang to Suzhou route through Shanghai first, which adds about an hour to any such trip. I'd estimate that the trip by train will take you at least 6-7 hours each way; less if there's a new CRH high-speed train serving this route. (I wouldn't count on this; southern Zhejiang is much further down the "pecking order" than the north down through Hangzhou and Ningbo) There may well not be a large choice of train times available to you either direction. You might also be able to get a bus, but would not be surprised if you had to change buses in somewhere like Hangzhou in the process. This will also be a long, uncomfortable trip. Buses operate pretty strictly only during daylight hours, so again you won't likely have a lot of choices. So you CAN get to Suzhou within a day, but you won't be casually zipping up here for a day or for too many weekends. Your Saturday nights at the Shamrock will likely be tempered by an early train or bus home on Sunday.