Anu- in this context as I understand it just means continuously or a kind of repeated action, at least as I've heard it explained.
Anuvitakketi is the verb for continuously thinking, directing the mind, etc. about the topic, which matches with the meaning of anu- and the context in which it occurs. Similar to anussati, etc.
Vitakketi, as others said, is simply the verb meaning 'to think,' and 'vitakka' is the noun meaning 'thinking'.
Vicarati can also literally mean 'wander', but also refers to pondering/examining/deliberating over (as in a kind of 'mental wandering') over a topic, such as in this context, or in the context of vitakka-vicāra (with the verbal noun vicāra once again, "wandering" or "pondering" or "examining" as in the dictionaries).
The definition of pavicarati in some dictionaries is "to investigate thoroughly"— the pa- contributing the "thorough" aspect. Here is part of the definition:
often emphasising the action as carried on in a marked degree
The prefixes (in this case) do not change the core meaning of vitakka/vitakketi and vicara/vicarati. They simply add some natural. contextual intensification to the words, as is very common in Pāḷi speech. It's good to remember that, despite what later Abhidhamma ended up doing, Pāḷi was not a highly technical, Buddhist-only language that can express meditative concepts with extreme precision unlike other languages. It's a normal, average Middle-Indo-Aryan language, distantly related to English, with normal words and normal meanings. Some of those things end up having more specific usages in Buddhist contexts, of course. But vitakka and vicāra are not one of those as used in the suttas and from the lips of the Buddha.
It can be good to put ourselves in the place of the audience. We all speak English, so let's so we're in an English speaking environment, and pretend we're all native English speakers. And a Buddha comes to teach us Dhamma, and he says:
"When you're in this meditative state, there will be thinking and pondering, and joy and happiness."
How would we interpret that? Would we say: "By thinking and pondering, he must mean a mental application that directs the mind to a nimitta and then sustains its focus on a conceptual object"? Of course not. This is the same position that the people listening to the Buddha were in. He was speaking a language that they all knew since birth, and that they used in everyday life for completely normal daily tasks. Words such as thinking and pondering/examining were used all the time, I'd presume, considering people still thought back then and there are no other words in the language.
Even if the Buddha wasn't speaking Pāḷi, it was a language extremely close to it, essentially a dialectal difference.
Also, recall that the Buddha would very specifically and explicitly re-define terms "in this discipline" when he used them differently. Such as when he defined the world as the six senses, or when he defined blood as breast-milk in some contexts, etc. etc. It happened all the time, and he would always explain himself. Why? Again, people were using this very normal, common language. If he was using a term strangely, he would need to specify, just as any English (or any other language) speaker would need to do when they use a word differently. He never did that with vitakka and vicara, and these come up thousands of times in all sorts of contexts, the jhānas being one of the most frequent.
Mettā