Micromanaging your inventory is undoubtedly one of the most critical aspect of this game that players will constantly pay attention to. And with so many resources to collect and so little space to them in, this process can become an infuriating chore very quickly. Things do become easier as you gain more slots for your exosuit (and starship), but expect to spend the early-goings of your space adventure debating which item to keep or to throw away. Worse still, even though you are allowed to transfer items from your suit to your ship at any time, the amount of space the latter has is also really small.
So what is a space adventurer to do? Fortunately you can find exosuit slot upgrades inside Drop Pods that randomly spawn when you explore a planet's surface. To locate these, you must first find a signal scanner -- a device that shoots a red beam towards the sky. These objects are indicated as a point of interest ("?") when you use your suit scanner by pressing L3. Once you've found a signal scanner, you'll need a Bypass Chip (crafted using 10 Iron and 10 Platinum) to operate it. After which some options will appear. Pick "Shelter" and an icon will be added to your HUD on top of the screen.
Go to that location and you should find either a trading outpost or drop pod if luck is on your side. Suit upgrades can also sometimes be found locked away behind doors that require an Atlas Pass, so you can try looking there too if you can open these doors.
Now for your ship. There is unfortunately no way to upgrade your current ship using mined resources so what you have in it is all you're ever going to get. The only way to get more inventory space for it is to buy a new ship from traders that land on space stations (which obviously costs quite a fortune), or repair a damaged starship that you occasionally find while exploring planets. What's odd is that there will come a point in the game where your exosuit has more slots than your ship... Well at least that takes care of the inventory space problem.
Finally for the Multi-tool. The basic five slots you initially have in it is obviously not enough (what else is new?), if you are to install all the nifty techs that you find for it. And just like your ship, the only way to get more slots is to purchase a new multi-tool. The trading terminal in space stations is as far as I know, the only place you can buy these. They will cost quite a bit so hopefully you have plenty of cash to spare.