When do I get to upgrade the ship

In the top-right of your screen, there is a button to teleport back to the ship. It's right above the Quests button.

Note that it will only be available while on the surface of the planet, outdoors. Going underground will make it vanish, like this:

Note that the definition of "underground" is a bit crude. All you need to have teleporting available is that:

• There is no background wall on the square your head is occupying
• You haven't gone too far down.

This means that, for example, in the following image, I am able to teleport:

Even though I'm underground, with ceilings and background everywhere, because I'm not too far down, and because I've punched out the one square of background where my head is, teleporting is possible.

Lastly, while you are onboard your ship, the same button will change to allow you to teleport back down to the surface at any time, without having to actually walk over to the teleporter. It's not much, but it's handy.

1.0 and later

If you are playing on the "casual" difficulty, you can use the button to beam up to your ship anytime, except when you are in dungeons. This means even on Outpost, where on older versions you were obligated to use the teleporters to get back to your ship. If you're playing on the "survival" or higher difficulties, the functionality is the same as described earlier.

As of the current Beta v. Enraged Koala, the only tools that can be directly upgraded are pickaxes, and their cousins, drills. The Stone Axe is the only woodcutting tool available in the game, though a chainsaw does exist in the game's files (but is unobtainable).

There are, however, a number of tool-related mods which might have something, if you want. And, there's always the possibility of just about anything being added in an update, including that chainsaw finally becoming obtainable.