Reading books and checking websites has/have helped them

The second is correct in this instance. Because there are two subjects joined by and, the subject is plural and requires a plural verb. Gregg's Reference Manual (11th ed.) paragraph 1002a explains this as follows:

If the subject consists of two or more words that are connected by and or by both ... and, the subject is plural and requires a plural verb.

Think of ‘reading books and checking websites’ as being ‘an activity’. A completed ‘activity’.

‘Something you did’, which collectively is ‘activity’. This makes it, in English ‘one thing’ (implied).

So we would use ‘has’, which refers to the singular. ‘Have’, in my view, is absolutely not correct in this sentence and sounds very weird to my (English!) ears - iffff... those sentences are complete.

Howeverrr...

...If you continued on with the second sentence - for example:

‘Reading books and checking websites have helped them to gain a further understanding of the Italian Renaissance’

Then in that case, we do need to say ‘have’. Because now, what is implied is ‘the activities’. Multiple and discrete ‘activities’. Which then takes the plural.

Whyyyy?

• Single things (even when ‘implied’) take the singular ‘has’

‘This completed activity or group of activities’ (perceived as one thing - ‘this activity’ - singular) has helped them

• Multiple things (even when implied) take the plural ‘have’

‘These two activities’ (perceived as multiple things) have helped them

In a nutshell: use ‘has’ when you want to refer to the individual activities that took place. Use ‘have’ when you want to refer to ‘the activity’ (or ‘effort) that ‘went on’.