Now that you are well situated at home, it's time to Strike Out! Your goal for doing this is to map out the world to find resources, find the Maxwell Door, and find the Five Things that will allow you to move to the next world. There are a never ending number of worlds so leaving this world does not mean that you've "won". It merely provides you access to a new world with more resources in case you start to run out though some people consider moving to each new world to be the goal of the current world.
First, some things that you may want to try to get together before setting out: Walking Cane, Beefalo Hat, Bush Hat, Jerky or Meatballs, fire building materials, a Log Suit or Marble Suit, Football Helmet, Spear or Tentacle Spike, Ice Staff or Pan Flute or Sleep Darts. None of these are necessary but they will all help you survive the creatures out there and stay out searching longer than you could otherwise. For fire building materials, it is ok to bring rock if you have extra and build Fire Pits instead of Camp Fires as this will allow you to create small safe havens around the map. THis is particularly useful if you plan to stay in this world as long as possible.
Finally, there's one thing you really should make before leaving if at all possible: The Divining Rod. Unfortunately, it requires Gears which can only be found on a Chess/Maxwell Biome by killing Clockwork Knights and Clockwork Bishops. If you have some of these, then the other ingredient you are going to need is Nightmare Fuel of which you'll need 4.
The easiest way to get Nightmare Fuel is by killing Beardlings which are ultimately just the form that Rabbits take after you lose about 60% of your Sanity. There are of course several ways of going about this such as eating raw Morsels or jumping through Worm Holes several times. Before doing this, it's also recommended that you come up with a way of getting your Sanity back when you are done. If you've been following along then you should have plenty of straw so the easiest method is probably just to make up a Straw Roll and have extra food nearby to counter the Hunger penalty from sleeping through the night.
When you are ready, clear out most of your inventory and fill it with recently trapped rabbits (don't kill them yet). Do whatever you need to and lower your sanity. Once the Rabbits in your inventory (or field) turn into Beardlings, start "Murdering" them. Keep in mind that this will still create naughtiness points so be on the lookout for a Krampus. Each killed Beardling has a chance of dropping either Beard Hair (the only way of getting it if you aren't playing as Wilson) or Nightmare Fuel. When you've gotten enough Nightmare Fuel or killed all of the Beardlings, use your Straw Roll to sleep it off. This should get your Sanity up a decent amount.
If you have the ingredients, make your Divining Rod but if not, you'll just have to wander around for now until you come across the previously mentioned Clockwork enemies. When you do, you'll likely want to collect extra Gears if you can because this will allow you to make a new Divining Rod after you travel to a new world.
There are generally two ways of exploring the world, each with its own merits. You can start on a road and see where the road takes you or you can find a coast and start mapping out the coastline. From there, you can search more interesting areas as you see fit. The method that you use is really depending on where you have the Divining Rod or not.
Without the Divining Rod, your goal is to find the Maxwell Biome and the best way to do that is to use the roads. There are two reasons for this: 1) Roads will take you quickly from one land area to the other which will help you find new Biomes more quickly than searching the coast. 2) The Maxwell Biome generally isn't up against the coast line but closer to the middle of a land mass which is also where roads tend to be.
Once you've built your Divining Rod, you'll likely want to stick to the coasts. This isn't to say that the roads are bad because it is definitely a viable strategy but you'll be more thorough using the coastline instead. The Divining Rod has a rather large range and can cover most medium distances well so by sticking to the coastlines, you'll likely cover most of the land further in as well. But you're also more likely to find bridges to other land masses that you couldn't access before and finding these is usually key to finding the Five Things since they tend to be well spaced out.
Regardless of your end goals, right now you want to find the Box Thing, Crank Thing, Potato Thing, Ring Thing and Wooden Thing so that you can leave if/when you want to. After that, you can do whatever you like in this world. Once you have your exit, you've got the ultimate freedom to do as you want with this world because you'll be able to leave it for another any time you wish. Keep in mind that once you leave, you'll never be able to come back so it is wise to be prepared before you go.