The first technology you need in the Exploration age is Cartography, which allows all civilian and support units to enter deep ocean tiles. They take up to 20 damage per deep ocean tile, but on a continents+ map you should be able to travel not more than 5 deep ocean tiles and make it to the coastal tiles of the islands between the continents. Note that scouting in Civ7 is extremely powerful, so once you have the Imago Mundis memento, you can already find good pathways to the islands in the Antiquity age by using the scout's Search action on the eastern or western shores of the starting continent. On a map without islands, it might be possible to overcome the deep ocean early by researching astronomy and researching a fleet commander. You can then go 5 deep ocean tiles with the fleet commander, which will damage him, but not the ships he has packed, and then unpack and do the rest of the way with the still undamaged ships. You just need to hope that the new world isn't more than 10 tiles away. :) And then you'll need to find an independent in the new world and use diplomacy to turn him into a city, as you can't cross a large gap with anything but ships.

A regular infantry, cavalry, or ranged unit cannot enter deep ocean tiles until you have the mastery extension of Shipbuilding researched, which is half way down the research tree. However, an army commander with 4 infantry, cavalry, and ranged units packed inside counts as a support unit, and only needs Cartography to cross deep oceans. So an early military conquest of the new world is possible, if there aren't more than 5 deep ocean tiles to cross. Still, being good at science and getting shipbuilding as fast as possible is a huge bonus for the Exploration age.

The other big new factor in the Exploration age is religion. You will want to rush to the Piety civic as fast as possible and found a religion. Then you can build temples, which allow you to build missionaries, which allow you to convert cities. If you send a missionary to any built-up district in a city (yours or others) that doesn't have any religion yet, he will spend one of his charges to convert both the urban and the rural population to your faith. If the city already has a religion, you will need to convert twice: Once in a rural district, and once in an urban district. There are two little round icons under the city name telling you the conversion status, but the game isn't explaining all this very well.

When founding a religion early, I think that the founder belief "Interfaith Dialogue" is very powerful, as it gives you +4 science for every foreign settlement following your religion. This is hard to follow, as the UI is horrible, and on the starting continent you might need to constantly send out new missionaries to keep settlements converted to your side. But the science boost helps with getting to shipbuilding early, and nearly all age progress is tied to that.

If you dislike religion, you can somewhat ignore it, but as this is called the Exploration age, you can't ignore the exploration of the second continent. The units you want to send to the new world early are thus ships (you start with one cog), scouts, settlers, missionaries, and army commanders.And you will want to do all this as early and as aggressively as possible. If you wait until shipbuilding, the age will be half over before you can even start gathering your first legacy point.