I can pronounce the english "r" properly in most words but it sounds closer to a "w" when it follows a "g" and sometimes "k" sound. The words that are the most problematic for me are agree and agreed. When I normally say "r" I pull my tongue back and touch the teeth with the sides of the tongue. Should I point my tongue tip up? And how can I practice this?
The "r" sound in American English is often described as a postalveolar approximant (/ɹ̠/) and not a trill (/r/). This means that rather than producing a vibration or trill, air is constricted without vibration. The tongue in this postalveolar position is behind the alveolar ridge, or the left-to-right ridge you feel when you lift your tongue up in the mouth. The tongue doesn't have to touch the roof of the mouth to make the sound, but it is raised. The lips often round when producing this sound, especially at the start of a word or following a consonant.
The /w/ sound you are producing is an example of rhotacism, or difficulty pronouncing "r." /w/ is a labio-velar approximant; "velar" means that the tongue lifts up further to the back of the mouth, and /w/ also requires rounding of the lips. The difference between /w/ and /ɹ̠/ (or similar possibilities like /ʋ/, a labiodental approximant) is small; exaggerating a "ruh ruh ruh" and "wuh wuh wuh" can help feel the difference.
I would work on "r" with a mirror (which lets you observe lip movements) and a series of set syllables ("dra" from "dragon," "gra" from "grab," and so on) that are practiced, occasionally with reference to YouTube videos or other resources on pronouncing "r." In other words, start with easier syllables that allow you to feel where the tongue is in the mouth, and then scaffold upward into full words and phrases.
Instead of trying to say "agree" normally, try practicing by saying "ag Ree" as two separate words very slowly a bunch of times. Then practice it more quickly, eventually letting the two parts start to slur together.
In US English, anyway, it's a normal "r" sound, with the back of the tongue pulled back in the throat. The tip of my tongue is no different when I say "great" than when I say "rate".
If you are OK producing the "r" at the end of a word like "tiger", try saying "tiger ate" a number of times. Try to merge "tiger" and "ate" and then eliminate the vowel between the "g" and the "r" while you're doing it, until you're saying something more like "tie great".
Then, speaking of tigers, it may also help to watch and mimic this video of a tiger saying "Grrrreat!" (Tony the Tiger from the Kellogg's Frosted Flakes commercials.)
With a few exceptions, consonants followed by consonants are "sucked" in a little bit.
"St" as in "stout," when pronounced slowly, can be broken down into "su-Tuh." Similarly, the "ou" can be pronounced "awe-oo," where "awe" is the short o an "oo" is the long u.
"Str" as in "street," would in the same vein, be pronounced "su-tu-ree-t."
Sounds such as "ch" and "th" don't follow this rule.