Yes, you can get to max level by just following side-quests and main story lines, mostly.
I leveled my first job from 1 to 50 without any grinding. Just do every sidequest, every dungeon at least once, join FATE as it happens, no need to grind.

Just some hint for you, in case you miss any:
- In "Character Configuration" > "General" > "Duty Recommendation", turn ON for both option. This will alert you all the quests available in an area.
- Seems like a lot of people keep missing those sidequests around lv46-48. Those are in Waking Sand, easy to miss.
- If you go to FFXIV official website, there's a online library which you can search for all quest by level range, use that to see if you miss any.

This forum is filled with a lot of negativeness, but there are a lot of people enjoying the game thus never post
Enjoy!