1.1 What is aspiration?
When a voiceless stop and a following vowel (or liquid consonant) are pronounced in Spanish, during the closure of the stop there is no vocal fold vibration. The vowel on the other hand, is fully voiced. This means that the release of the oral occlusion must be coordinated with the activity of the vocal folds, so the onset of voicing starts at the moment of the release, or very shortly after it. In English, however, there is a considerable “gap” between the release of the occlusion and the start of vocal fold vibration. The result is a voiceless aspirated stop, so we hear a small [h] sound between the release of the stop and the vowel. This allophone is typical in utterance-initial position unless the voiceless stop is preceded by an /s/ as in spill, still. In these cases, there is no aspiration in English either. So the closest production of the Spanish voiceless stop will occur in English words such as spill, spot, still, but without the [s].
Note that Spanish /t d/ are more front than their English counterparts.