What are the core Mahayana and Vajrayana Sutras? When starting a more in-depth study of the Dharma from a Mahayana and Vajrayana perspective, what are the core (3 most recommended) Sutras (or texts), and the reasons why they are considered important, to have on your reading list in each of those traditions?
I think your question is much larger than can be answered with just three texts. Mahāyāna is not one thing, it is many Prajñāpāramitā, Madhyamaka, Pure Land, Yogacāra, Chan/Zen, Huayen etc. Similarly Vajrayāna is not one thing either. 

For the Mahāyāna a comprehensive guide would have dozens of texts. At a minimum:


Aṣṭasāhasrikā Prajñāpāramitā Sūtra  
Mulamadhyamikākārikā śastra  
Abhidharmakośabhāsya śāstra  
Yogacārabhumi Sāstra
Sukhāvativyūha Sūtra (x2)  
Sūtra of Huineng  
Saddharmapuṇḍarikā Sūtra  
Avataṃsaka Sūtra   


For Vajrayāna the situation is different. Tantras cannot be read on their own. For Shingon the two main texts are


Mahāvairocana Sutra (aka Mahāvairocana Abhisaṃbodhi Tantra)  
Vajraśekhara Sutra (aka Sarvatathāgata-tattvasaṃgraha)


A good introduction can be found in Kūkai: Major Works by Yoshito Hakeda. Tibetan Tantra includes many more texts, such as:


Guhyasamāja Tantra  
Hevajra Tantra  
Kālacakra Tantra  


But most teaching is done through commentaries. Many of these texts cannot be understood without a commentary. Some of them require specific initiations.