Main ways to watch Filipino series is through television or online (through the web / over-the-top media services). To better explain, let me categorized them into: Teleseryes and Filipino Web Series.

1. Philippine television drama, also known as Teleserye — is a form of melodramatic, serialized, televised fiction in the Philippines. Teleserye is derived from two Filipino words: "tele", short for "telebisyón" (television), and "serye" (series).
Teleseryes share characteristics with and have roots similar to soap operas and telenovelas. They have evolved into a genre with unique characteristics, however, and often reflect Filipino social reality. Teleseryes are aired in the afternoon and prime time, five days a week. Their audience crosses age and gender lines, and they have the highest advertising rates in the Philippine television industry. Series last from three months to a year or longer, depending on ratings.
Philippine TV dramas also include serials and anthologies, usually shown weekly. These dramas have a finite number of episodes and usually last one season, again depending on ratings. [*]
Main TV Networks these teleseryes were and are being aired are: ABS-CBN, GMA, and TV5.

2. Filipino or Philippine Web Series — Shows released online or through over-the-top media services. They are shorter in episode length compared to the traditional teleseryes. They usually cater to younger audience who are tech savvy.

Note: Since times are changing, many shows are released both on television and online. The earliest notable one was ABS-CBN’s On The Wings of Love (2015-2016).

WHERE TO WATCH FILIPINO SERIES ONLINE?
1. On-The-Top Media Services - iWantTFC (ABS-CBN’s OTT), VivaOne (Viva Entertainment’s OTT), Netflix, Amazon Prime, Viu, HBO Go, WeTV, iQiyi, iflix, etc.
2. Youtube Accounts of Major TV Networks - ABS-CBN, Jeepney TV (archive of ABS-CBN), GMA, TV5
3. Free websites

Next Post will be recommendations! I’ll add the links once MDL let this new account.