Just in case it's not obvious, I'm not a computer expert. I've tested out all the information below on my own computer with no negative effects that I can discern as well as scanned all the linked files with both Malwarebytes and AVAST Antivirus software with nothing detected at the time of writing this. That said... I recommend you scan the files yourself and proceed at your own risk.
The first step is to download the base game file installer v2.02 as well as the v2.04 patch hosted over on the trekcore website linked here. There are two additional downloads (a ship counter and romulan pack) but in my installation both were included in the 2.04 patch and are unnecessary.
Unzip the downloaded files and install the v2.02 game making sure to note the file directory that you'll be installing it into. Next, do the same with the v2.05 patch making sure to specify the same directory as the base game. The screenshot below shows as a reference the default one that I used.
If you try running the game now in Windows 10, you'll likely get a "transpicture.ocx" error. If you do, you'll need to download the transpicture.ocx file at the webpage linked here. As with the above, make sure to run your virus protection software to make sure that the file hasn't been corrupted in the time between my checking it and your downloading it. Place the file in your c:\windows\system32 directory as shown below. (Editor's Note: Possible alternative methods are listed at the end of the post!)
You'll now need to register that ocx file in order for windows to recognize it. I had to download from Microsoft and install a Visual Basic run time program from the link here. Again, this is from the Microsoft website but always run a virus check just in case as it is an exe file.
I promise we're getting close! Next, you'll have to open a command prompt as an administrator by typing "command prompt" in your Windows 10 search bar. It should be the first result on the top and you need to right then left click on it to "run as Administrator".
That opens up the old DOS style command prompt window and you need to type "regsvr32.exe transpicture.ocx" into it and press enter in order to register the OCX file. In my case, it defaulted to the same C:\windows\system32 directory listed above where I put the OCX file. You should get a successful confirmation after doing so if it worked.
Finally, right then left click on the shortcut for the game on your desktop run the "troubleshoot compatibility" function. On my computer, it selected running it under Windows 98 Service Pack 3 and the game hasn't fully crashed once.
You can now play the game! Occasionally, the program will hang after the enemy AI's turn but I've found that the quick fix is to do one or all of the following in this order: press escape, click on tactical if it is showing and the press escape to return back, or press the windows button then reenter the game window and press escape.
I hope this how to guide helps folks rediscover this classic game from the 1980's. I don't harbor any pretext that it'll be popular with younger gamers as it is a game design relic of its time but hopefully it'll rekindle some interest with folks nostalgic for the original.
10/2022 Edit: Thanks to Dr Kanukie for researching updated methods to installing this as programs that worked for me in 2018 may have changed in the years since. I haven't tested these out yet myself as I haven't gotten a new computer since so all the usual caveats already noted above apply. Dr Kanukie got the process to work by for 64bit systems like Windows 10 by putting TransPicture.ocx in SysWOW64 folder instead of the System32 one and using Windows XP SP3 compatibility settings instead of Window 98.
4/2022 Edit: Sometimes for no explanation Microsoft seems to be changing the availability of certain files. For now, the microsoft link above for the Msvbvm50.exe as shown in the picture does work. I myself haven't tested the Visual Basic 6 runtime pack that replaced the original link (here) but hopefully the direct exe file link above will continue to work fine. If the VB6 link alternative doesn't work, Tycen suggested in the comments below that it's also available at the following: vb6extendedruntime As always, run virus checks first before installing anything as I haven't tested these last two methods myself.