If you don't like spoilers, don't worry too much about the achievements on your first playthrough. Some are missable, but they are easy to mop up afterwards. Except: Make sure to collect all the flags in every episode (Flag Waver)!

• Make sure to collect all the flags in every episode (Flag Waver).
  
• Collect all the magnifying glasses in one of the episodes (No Sh*t Sherlock).
  
• Do one episode in one sitting without loading (Marathon Man)
  
• Try to kill a lot of human enemies and collect their tombstone (Grave Robber).
  
• In general: try to collect as many bonus items as you can (Banking Agent, Fresh Breath) and catch the falling coin bags (Raining Gold).
  
• Save after about every 4 levels in episode 1 and 4, so it will be easy to jump to a specific level afterwards (preferably on Medium).

Some stuff to look out for if you don't mind a little bit more spoilers after the picture.

The background lighting in this game makes no sense...

• An opportunity to kill an enemy with the dynamite at the end of a level.
  
• Try to get confused and slowed down at the same time.
  
• A pickaxe.
  
• A hidden Dopefish behind some breakable blocks.

You will have about half of the achievements already, look in the next section for some tips on the ones that are not obvious. Workshop levels do not count for any of the achievements as far as I could tell. Resetting a level or dying usually removes the progress you accumulated in that level, so if you're grinding something use save and restore.

This ancient website has pictures of all the maps of the original game: http://sfprod.shikadi.net.s3-website-us-west-2.amazonaws.com/games/secretagent.htm#maps. Also apparently the little dog enemy introduced in the new episode 4 was based on unused sprites in the original game :)

Some tips / clarifications for the less obvious achievements. (Most of the hidden ones have no description beyond "..." after unlock.)

Agent Swift
Finish any level in under 60 seconds

A good level to do this on is E1L05: the second level in the top right of episode 1.

Stack-em-up
Stack two barrels, one on another

Thanks to Caedus:
E1L11 works well for the Stack-em-up achievement: just put the right bottom barrel all the way to the left, right under the ledge and then get the second one from up.
Thanks to D13GO_{HoF}:
In E1L03 (below the teleport) you can stack both barrels, but you may need to restart the level afterwards.

Key Master
Hold 6 keys in your inventory at once

Easy in E3L01 (I think it's on the teleporter island), the level with many clouds, thanks JoesGuy.
(Back in the 90's version, it wasn't possible to have more than one of the same color key in your inventory... that would've made this level a lot more frustrating challenging)

Arachnophobic
Destroy 10 mecha-spiders

These are the spiders that periodically turn invisible in episode 4. A good level to find them is E4L15, in the top right of the overworld.

Destruction Man
Blow up enemy with a Dynamite

This is the level-ending dynamite you place at the door, can be tricky to get the timing right.
Thanks to Caedus: E1L13 is a good level to try.

Bad luck!
Get confused and slowed down at the same time.

Can be done in E1L15, an open air level with a single red car in the beginning (thanks Torchfire). I'm not sure where this level is, it may be the one on the transporter island.
Even easier in E3L15 (thanks D13GO_{HoF}). This is the level at the bottom right, reachable when you start the episode.

Better than Bond
Complete any episode without taking damage.

You can restart each level as often as you like, just make sure that each time you finish you have all lives. No save scumming needed. Episode 1 seems like the obvious choice to try.

Coffee Jacuzzi
Life is better in the hot tub.

The clue is in the title.
Stand behind a coffee mug for 15 seconds, much like the achievement icon.

Crawling no More
Destroy crawler bot while in the air on medium or hard difficulty.

This can be done in E4L09, the pyramid level in episode four. This level is just south of your starting position. You need to destroy the bot while it is in the air. Credits to Cain and Zuko!

Originally posted by Zuko:

This took me a few attempts - on medium the crawler bot has to be shot twice. So what you need to do is shoot it while it's still attached to the ceiling - this will cause it to drop where you can shoot it again and get the achievement.

Agility
E4L10 without killing any enemies and with a perfect health.

This is the second unskippable level in episode 4 (open the menu to see the level number). You can do it on easy, but that won't help you much. This will also give you Espionage (Complete any level without killing any enemies).

Swim, swim, hungry
...

Clue 1: Find the hidden Dopefish[quake.fandom.com]
Clue 2: It's hidden behind a 2x2 set of breakable blocks.
Clue 3: In episode 4.
Clue 4: Level E4L12.
Clue 5:

Treasure!!!
...

In the overworld of any episode, after completing all 16 levels but before escaping, jump over the shark when it flashes (see below). A bunch of money bags will spawn over a period of a couple of seconds all around the island(s). Collect them all to get the achievement.
As you collect the first one, a timer of 45 seconds starts. So save before you try and pick a smart starting bag to start a quick route. I found episode 4 much easier than episode 1.

Jump the Shark
...

There is a specific spot in each episode overworld where you can jump over a short bit of water by pressing the jump button (direction doesn't matter). If you do this when all 16 levels are completed and time it with the shark you get the achievement and set up the achievement Treasure!!! (see above). To get the timing correct, jump when the shark starts flashing.
In episode 1 the spot to jump is from the little island with the teleporter.

X Marks the Spot
...

Clue 1: You'll know in which episode this is when you get to it.
Clue 2: It's about the island in episode 4, but how to get there?
Clue 3: You will need to pickup a pickaxe from inside some level.
Clue 4: It's in level E4L13, at the top right part of the overworld.
Clue 5:
Route: Take the top left teleporter, then the one with the S on top. Shoot the tube and pick up the pickaxe. Don't need to finish the level afterwards. That opens up a path to the island with the X on the bottom right in the overworld. If you go there you'll get to a very difficult hidden bonus level. You'll get the achievement immediately, but good luck!

Rokket Mushroom!?
...

You can get this powerup in the final boss fight of episode 4. After killing the boss (press fire at the console in the middle), get both the scuba gear and the jetpack and dive all the way to the bottom left.
It will unlock the CGA mode in the options.