Linguistically, it isn't particularly helpful or entirely valid to only consider either vowels or consonants independently without any consideration of the context in which they appear. (Yes, some may argue that the IPA is a list of distinctive sounds that occur in natural language, but there is a huge section of diacritics that alter how those sounds are made.)
There is an important concept in language (whether learning, using, creating, &.) that morphs our pronunciation. It is called Coarticulation. There are two main kinds of coarticulation: anticipatory and carryover/perseverative. This is something that we cannot escape from in language. The central concept behind coarticulation is that we cannot isolate individual sounds from their environments. We either prepare for the next sounds (anticipatory) or characteristics of one sound are carried over to the next (perseverative).
The sounds that precede another sound (usually a consonant) are affected by those following sounds (either in the same word or in the next word, known as liason). In this case, we prepare for the next sound by modifying the current sound. (One of my linguistics professors said this was the most common form of coarticulation, but I cannot remember the source he quoted.)
An example. In English, take the words "high" and "hue". Notice the position of your mouth when saying "high". The initial /h/ is open and unobstructed because it is preparing for the open unrounded /ai/ that is to follow. But when you say "hue", your tongue comes up to the top of your mouth and prepares for the /j/ (y-sound) that comes after it. Which one of these "h"s is the real "h"? They both are, but their pronunciation is affected by preparing for the subsequent sounds.
Though English has one of the richest vowel systems (and its vowels contain the vowels of most other languages), it does not have all of the consonants that would change how those vowels are pronounced (advanced tongue root, retracted tongue root, centralized, &c), nor does it have all the features of other languages that are applied to vowels or consonants (like breathy voice, creaky voice, rhoticity [English actually has this one], nasalization, more/less rounding, &c.). This makes the topic of vowels and consonants very difficult across languages.
It is ultimately wrong to only consider vowels and consonants as being "fixed" or "static" because they change so much across cultures, regions, languages, and dialect. Fully embracing and learning to pronounce all of the "vowels" and "consonants" of a language will lead to you naturally utilizing coarticulation to morph the sound into the correct one. In Chinese, when I learned the palatalized and retroflexed consonants as well as the /y/ sound, I realized that I naturally started pronouncing the words better and the Taiwanese people noticed. So, learn the sounds independently, yes. But practice them together with other sounds and you will find it helps your overall pronunciation.