Resources

For Tâi-gí speakers, 

those sites could help you pick up writing and reading Tâi-gí within in a week. (Within 3 days if you work on it, 8 hours if you work really hard.)

台語文讀寫速成法 - 潘科文台語理想國
學習台語文讀寫 基本的網路資源 - 潘科文台語理想國

雙倒水別莊 - 正手邊ê足濟連結資料會當參考
台語文數位典藏資料庫
台話字台語文
陳冠學先生的台語資料

For non Tâi-gí speakers,

Forumosa great resource

listening to Tâi-gí might be a good start so you get a good sense of what the language sounds like.
世界台 Sè-kài-tâi  - Weekly international news in Tâi-gí. (Sè-Kài-Tâi Facebook)

https://www.reddit.com/r/ohtaigi/ - Online Forum, offers many helpful links on the right hand side column, such as Getting Started and Dictionaries, etc.

Other earlier sites:
Pháiⁿ-sè and Tailingua by Michael Cannings

News in Tâi-gí

As of Sep. 2017, I found daily News in Tâi-gí you can watch online:
(It's referring to Taiwan time zone.)
Broadcasting time may vary from time to time, please check their website for up-to-date program.

8am - CCT 台視台語新聞(台視HD)
10am - FTV 民視台語新聞 (民視新聞台) youtube streaming
11am - CCT 台視台語新聞(台視新聞台) youtube streaming  (CCT's Taigi News playlist, updated daily)
noon - PTS 公視中晝新聞 youtube live streaming (Best quality news, in my opinion)
2pm - NextTV 壹電視 app streaming iOS Android
3pm - FTV 民視台語新聞 (民視台灣台) youtube streaming
6:30pm - PTS 公視暗時新聞  youtube live streaming

**bonus: A program @ Chengte (religious/Buddhism) channel that teach English in Tâi-gí, quite interesting.  Same program replays 4 times a day.  Timetable here.  Youtube live streaming.

Informative Facebook pages:

taigikho - 失控的台語課
itaigi - iTaigi

Happy learning!!