The first thing to know is that three of the achievements are now broken and will never register as completed. These are the two introductory ones "Your First Visitor!" and "Earned 1000 Dollars", along with "Reached the 13th Floor!" which was originally awarded for doing just that, but at a certain point the developer reduced the number of vertical levels available from 13 to 5 without getting around to updating the achievement criteria, so it's now just impossible. The best you can do as an achievement hunter now is 23 / 26 (88%).

First of all, do yourself an enormous favor by going into the settings menu and turning off asteroids. While the asteroids provide one of the only real challenges in the game (such as it is), they are not a balanced or particularly survivable challenge, and you don't need them turned on to achieve any badge, so get rid of them.

Eventually you'll want to turn off random events too, but several of the easier achievements are related to those, so leave them on for now. In particular, two are obtained by giving both possible answers to the alien who wants to give you pamphlets ("Accepted Alien Literature" and "Rejected Alien Literature"), with no other apparent effect from the choice. He'll ask again soon no matter what you choose to do with the first offer.

Another two badges ("The Saviour" and "No Room Here") are given after you reply to the captain of the ship who asks to transport over a huge crowd of refugees (usually several times the station's current population), though he will still drop all of them on you no matter what answer you give him. This will usually result in a sudden massive overload of food and particularly oxygen resources, followed shortly by mass death, but it can often get you "FIFTY TOURISTS AT ONCE!" during the brief period where they're still alive, and when they all die that counts towards your "Mass Murderer" and "GENOCIDE!" badges too.

To research things, you need a scientist, and to get access to the science menu you need a station commander. This means you need at least 2 permanent residents to deputize onto your staff. A permanent resident happens when you have constructed a room at least 2x1 in size and designated it a dark green "Personal" room. If such a room is available, a tourist who is particularly happy with the station's services may choose to move into it, and a bed will appear on the floor somewhere in it. At this point the alien has become a Permanent Resident (there are 4 achievements related to getting up to 20 of these) and it can be assigned one of three jobs: Commander, Scientist, or (if Capitalism has been researched) Merchant. (They will also start paying you $100/day in room rent. If you have assigned them to the Merchant job and there is space in the Commercial area for them, they will also pay $50/day in taxes. All of this adds up nicely, and it's quickly possible to be earning $2000-3000/day net of daily robot costs.)

You'll also need a Command room, but it doesn't have to be very big, just 2x2. In it, place a command console and a science lab. More than one scientist can use a single lab to make research go quicker, though there is no further use for scientists after all 7 techs have been invented. This will quickly get you the "Egg Head" badge. At that point, just keep one Commander and switch everyone else to Merchant.

One of the biggest problems people find is that the robots often seem to eventually go on strike, no longer repairing or building things, even as it becomes more and more important that they hurry up and do it. Eventually this results in all your equipment blowing up. It seems like the bug may be related to building vertically. If you never place any stairs, they don't seem to stop working. So don't place stairs!