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I'm in doubt about what is more effective to have in this game:

1. More attacks (melee and/or shooting) without AP modifiers
2. Less attacks with AP modifiers

For example, a unit of Tzaangors (Thousand Sons) have options for a Tzaangor blade (-1 AP, additional attack) or a Chainsword (additional attack) + pistol.
This means I have 2 melee attacks with -1 AP OR 2 melee attacks without AP + shooting from the pistol (also no AP modifier).

I generally feel more towards the -1 AP because you have a better chance at wounding something. However with pistols you could potentially soften up your target before you charge (and have more purposes besides melee)...

What are your experiences in both cases? In general, better AP modifiers or just more dice throwing?

If I didn't make any mistake... the Tzaangor blades are better (for melee) (versus T4/3+).
Unit of 5 with Champion with +1 A ...
... with Tzaangor blades make 1,833 unsaved wounds (11 attacks)
... with Chainswords make 1,222 unsaved wounds (11 attacks) and 0,278 unsaved wounds with the pistols (5 attacks).

http://www.dice-hammer.com/

-1AP is better than +1A, most of the time. So I'd go with the blade.
TS has better shooting outside Tzaangors anyway

It depends on too many factors. In this case -1 AP is probably usually better.

Thanks for the responses guys.

Most of the time, when I'm list building (using points), I struggle between choosing a weapon with mediocre damage (say, auto cannon) or superior damage (lascannon), purely based on the amount of dice I can throw. I feel like I'm often rolling bad with dice so to counter that I search for ways to just throw more dice.

When I look at a Dreadnought fox example, I come up with a Lascannon/Stormbolter/Fist version where he can give support fire and do something in combat. Or is an Autocannon better in general? The close combat weapon is because 8th edition is easier to get into melee so I want to make sure I can fight back properly. I feel the need to perform on both ranged as melee. Is this the right approach overal?

It depends on the rest of your list. If your dreads are your main fire base units, then the 2 twin autocannons, or twin autocannon + twin lascannon are the best option. If they are just counter charge units that have some long range fire then your version is fine.

The right approach is to consider the role of each unit in your army and have it optimized to that role.