anfassen
to touch
To touch something with the hand or fingers.
Vorsicht, die Herdplatte ist noch heiß, nicht anfassen!
Careful, the stovetop is still hot—don't touch it!

to touch / tackle
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to touch
To touch something with the hand or fingers.
Vorsicht, die Herdplatte ist noch heiß, nicht anfassen!
Careful, the stovetop is still hot—don't touch it!

'Anfassen' (to touch / tackle) is a separable verb. The prefix 'an-' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Fass den Hund bitte nicht an, er ist sehr scheu.
Please don't touch the dog; it's very shy.
You can recognize the meaning from the object: With a concrete object or a person, it means 'to touch'. With an abstract noun like 'Problem' (problem) or 'Thema' (topic), it means 'to begin dealing with it'.
Darf ich den Stoff mal anfassen?
May I touch the fabric?
In the meaning of 'to begin a task', the verb is used metaphorically. You do not physically 'touch' the problem, but begin working on it. Typical objects are 'ein Problem' (a problem), 'eine Aufgabe' (a task), or 'ein Projekt' (a project).
Sie weiß nicht, an welcher Stelle sie das Projekt anfassen soll.
She doesn't know where she should start tackling the project.
When 'anfassen' (to touch) describes how someone handles a person, an adverb is almost always added. This adverb describes the manner of the touch, for example 'sanft anfassen' (to touch gently), 'vorsichtig anfassen' (to touch carefully), or 'grob anfassen' (to touch roughly).
Er fasste sie grob am Arm an.
He grabbed her roughly by the arm.
to begin
'beginnen' and 'anfassen' can both refer to starting a task. 'beginnen' only says that something begins or that you start doing something. In this figurative meaning, 'anfassen' is more about how you practically tackle a task or problem. It often sounds active: you take something on and work on it. For the normal start of an action, 'beginnen' is more suitable. When it is about dealing practically with a difficult task, 'anfassen' fits better.

Dieses Problem müssen wir vorsichtig anfassen, weil viele Leute betroffen sind.
We need to tackle this problem carefully because many people are affected.
Lisa beginnt heute mit dem neuen Bericht für die Arbeit.
Lisa is beginning the new report for work today.