Should work the same as in US - assuming you either get a local SIM card or have roaming packing. Be aware though that data roaming in most countries in Asia is very expensive, so often better to just get a local pre-paid SIM Usually need a passport to register the SIM), or get a package for roaming before you leave.
Performance of cell networks varies a lot in Asia - Korea, Japan and Singapore have 3G, 4G/LTE and will be very fast (Korea used to have their own networks and foreign phones did not work there, but it's OK now), Thailand and Malaysia have decent LTE/4G, India is OK in cities, poor in the countryside, Indonesia is patchy and Philippines is the worst in the known universe.
Free WIFI is common in Starbucks and in some airports, most hotels have (slow) WIFI.
From my own experience most Asian cities (except in Philippines) and most holiday resorts have better cell data reception than when I travel in US or UK. The final internet connection can be a bit slow (most connect via Singapore), so if you are accessing sites back home it can be a bit slow, but using 'Asian,' services like Line is fast, and Facebook etc have local servers.
China can be slow, but that's because of the 'Chinese wall' censorship system, and Facebook and some other Western sites are blocked in China.