Ik lees een boek op het strandlaken.

In Dutch, the present tense for ik is formed by removing -en from the infinitive and adding -s. The -t ending is reserved for hij/zij/u. So from lezen you get:
• ik lees
• hij/zij leest

Dutch requires an article with singular, countable nouns.

• een boek = “a book” (indefinite, any book)
• het boek = “the book” (definite, a specific book you have in mind)
  You cannot drop the article like in English.

Yes, it’s grammatically correct, but the nuance shifts:

• een strandlaken = “some beach towel,” any random towel
• het strandlaken = “the beach towel,” a specific towel (perhaps the one you brought)

Using het makes it clear you mean that particular towel.

strandlaken is a compound of strand + laken. The head of the compound is laken, which is a neuter noun and therefore takes het. In general, you need to learn Dutch nouns together with their article; patterns exist, but there’s no foolproof rule.

Dutch commonly joins nouns into compounds without spaces or hyphens. Here, strand + laken merge into strandlaken. Always check a dictionary for the correct, fixed spelling.

• op indicates location on a surface (“on”).
• in means “inside” (“in the book,” in het boek).
• aan often means “at/attached to” or vertical surfaces (“hanging on a hook,” aan de haak).
Since a towel is horizontal, you sit or lie op het strandlaken.

The default Dutch order is Subject–Verb–Object–Adverbial. If you front the adverbial/prepositional phrase, you must keep the finite verb in second position by inverting the subject and verb:
• Standard: Ik lees een boek op het strandlaken.
• Fronted PP: Op het strandlaken lees ik een boek.