Hi, I have just bought Farcry and wanna play it for the first time.
What mods should I install to enjoy the game?

IMO - none. FC is perfectly playable without any modifications and patches, especially if it's your first time with the game. I've never had to install anything to play vanilla.

Some solution to the enemy seeing through walls but GOG supposedly did do something about it.

The FarOut WideScreen mod.

Two main reasons:
1) It's the best (imo) fix for FOV and HUD on modern screen ratios;
2) It includes Silent Patch. Silent Patch restores the landmass reflections in water on modern OS's and enables Vsync (which only works properly with RTSS capping the framerate, but hey, we're lucky to finally have it at all).

Compatibility mods\wrappers and 16:10-16:9 support mod. PCGW article to look for (although as a wiki it never be 100% accurate info)

Also, GOG failed it to fix tents issue mentioned above. They use fanmade mod for that copied same mistake.
Devs make differ mechanic for it in latest MP-only patch. And put readme file how to restore. It is just text-editable (WordPad or HxD ofc, not standard Notepad) file.
And 3 values
- Non-transparent tents\some walls, not shooted throw.
- Shooted and entirely transparent - so AI will think there is no walls there\glass
- Shooted but not transparent - so AI will shoot only as back fire at player.

Mod used in current GOG version change from max-on to max-off instead of making middle possible. Stupid move next to stupid developers move. They already make it compatible with both MP and SP but chose MP-only realisation. Now in GOG it is SP only.
But you no need to download any mods for this - this one you can fix by WordPad yourself.

Such as?
And what are the compatibility issues (I'm not aware of any that would require a mod)?
_ When tinkering with .lua files, I used Notepad no problem.
_ I don't understand what you think the problem is. In GOG's FarCry, the AI doesn't see you through non-transparent obstacles, which is good. What does it have to do with multiplayer where AI-related things don't matter?

Ai dont see through tents cause with the fix tent walls are non-transparent entirely for them and cannot be shoot hrough.
That can be changed to "half-transpatent". Ai wont start shoot first but can return fire. And player can shoot enemy through tents (at random or using nightvision with bodies detection). Make noise - shoot or stone "grenade" - make sometimes that AI start to shoot through tent at you before you shoot them.

Half-transparent feature was implemented with multiplayer-only latest update to make it work in both way but for no reason argument set to entireltly transparent (no-wall) for AI in SP. Dvs even put explanation how that work but by default set it for MP only.
Current state is OK. Fix and mod based on revert it to very original form. But broke transparency (posssibily to shoot trough) for multiplayer mode. Thats what is patch made for and its reverted. Current fix made for singleplayer mode only. Ignore MP gameplay. Still change it how readme of lua file suggest better for realism and to make such mechanics work for both SP and MP.

DirectX 10 + WinVista+ and nowadays videondrivers broke few things. Mirrors in water (mountans and trees looks flying in water cause hills/ground not appear in the water).
Latest patfhes decrease VOD (detalisation distance). And game have support for 16:10 widescreen only. As mentioned in first answers here there fix-mods for that. Widescreen mod by ThirteenAG, SilentPatch. Not all og them fix everything, some include each others. Whichone most uptodate/allin - look at PCGW article (you can get there from your account GOG games library - FarCy - MORE - Support - additional resources at PCGW link).
PCGW as a wiki - not a perfect source. But still good to check. And good for subj game. List really needed fixes and properly describe what they do (and what not).
Well, except tent issue. Article just suggest ModDB linked mod as is. And GOG eid used it as is. Result - broken feature in multiplayer mode (possibility to shoot through some walls/tents). Although game already bundled with txt explanation how it can be changed/reverted - PCGW still dont have such information amd dont list all possible values/ways. Often issue on PCGW - suggest mods as is and incomplete info. But this time that affect only such tent-fix mod and good for SilentPatch/etc. Also SilentPatch really needed this time, for this game (look for more updated mod based on SilentPatch if there is such one - check their feature lists)

NotePad shouldnt be used for anything in non .txt and never for games files. What is shown in NotePad are not what is in the file (or what be saved to file after edits). Also NotePad put excessive info make it twice larger files. Some games can have filesizes validation and wont work with modified files where size differ or excessive symbols or if NotePad entirely change string order (for cfg and ini usually games dont care at arguments order there - you can add remove or edit them in any order, sometimes games will re-sort them by alphabetical next launch, sometimes they care for the order).

Universal solution is use HEX-editors to edit any text files as hex. That make it guaranteed exact symbols as you put there will be saved to file in exact section (addresses) of the file. Even changed 0<->1 in NotePad can change half of the non-txt text file or increase size in double. Free HEX editor is HxD. Often NotePad++ suggested as games text files editor.

WordPad although usually on forums not suggested (same as for NotePad people suggest NotePad++ or HxD instead) for mine experience it is entirely differ editor as NotePad and chanflges made by it dont increase filesizes while NotePad same edit do. But i did test it only for simplest edits (like 0->1 only in one string, for many edits still advanced text-editors are better)

Oh, I forgot about Silent Patch... I'm going to have to edit my earliest post in this thread.
_ My FarCry versions had 1920x1080 and 1280x720 options before I modded anything.

Since i dudint remeber clearly i didnt suggest exact solutions. And you still need such ingo. Go to PCGW for answers. There is everything youbneed - widescreen fix (fix fov/pov from 16:10 to 16:9), graphics fixes and imrovments. There is one complete/most uptodate fix-mod, i dobt remember which one of listed there most complete (but article should tell you that).
There is further improvments for VOD distance by console commands. But this one maybe incomplete info on PCGW and i found more commands somewhere else. It is not the single console argument (as i remember it is possible to use autoexec.cfg(or lua?) same as for Quake-based engines).

Alternate way is find 1.0/1.1 DVD edition or disc images and patch it to 1.2 x32 only. Or maximum is 1.32 x64 patches. 1.41 was for MP mostly broke GFX and those one tent issue.
I dont remeber if its possie to install two x64 patches to GOG version. If yes - you get alternate 1.32 x64 exe in Bin64 for a bit better SP experience (there still compatibility visual issues). English version only.
I cant guarantee this is prkper way to get x64 1.32. Poper install order are:
old 1.0/1.1 DVD editions - 1.32 x32 patch (or maybe even 1.2 first?) -- 1.32 x64 and AMD x64 content patch - 1.41 x32 patch. As result you get 1.41 Bin32 folder and 1.32 x64 Bin64 folder with all possible improvements to each of two version (incl additional maps for multiplayer included in x64 patches for x32 exe version).
Better create two separate profile (do not use deafult one! That always supposed to be create own new ingame profile) with two differ settings.
Run x32 for MP and run x64 for singleplayer.

You can also use ENBSeries modded dll by Boris ENB Vorontsov. When last time i test it it didnt fix compatibility issue. Maybe increase draw distance. And ofc ENB do not change widescreen support. If it did fix comp issues now as another alternate way you can use only ENB (start mod game with that) and for 16:9 support just play as is (it is even userul that FOV are closer) for 16.9 resolution or set some 16.10 - but lower than your display (lower than 1680x1050 for FHD display). And play with correct FOV but small black bars around - it is still wide resolution so they not be so big as for 4:3 resolution.
Of course videodrivers (+display settings, + games + wrappers/mods they all) should be set to Keep Aspect ratio (with integer scaling and forcing options checkboxes turn on if there is such options there).

If ENB did not fix water mirror compatibility issue - start with PCGW fix-mod suggested and at the end install ENB (General version or if there is FarCry specific version).

Man, if only I could understand what you're trying to say... Anyway, this thread is about mods that could be recommended to a first-time player, so we should keep it simple and I'm still going with FarOut WideScreen. The things you've mentioned are for enthusiasts only.

The water reflections and some other graphics are broken with some Nvidia drivers unless you do something about that. Hopefully easier to understand.

Specific GPU and OS?

Personal experiences? The problematic times were over ten years ago now. I find my memory full of holes and dare not swear it was this or that :-( I must have had twenty different GPUs over the years. I think I had an Nvidia GPU then.

----
I wonder it I am suffering from some real memory loss disease. Then it is time to leave these forums. Everything is so foggy.

winvista+ and nvidia drivers for long time (some before 400 series). For personal experience cant say for radeon current drivers. PCGW article say this is on all gpu and issue come from winvista+ (which is odd).
nvidia broke many old games with their drivers. maybe same for radeon.

What games?