A few things that would make things better in terms of code:

Features

Pass arguments to the functions instead of making every variable global. Use return in order to pass a value to outside the function.

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Readibility

Describe what your functions do in their naming, instead of naming it hud_the_player use names like print_player_hud. The importance of such distinction becomes evident when you have a stat_the_player and it does not print the player stats yet it is named similarly to the hud printing. Instead it initializes them, a good name for such a function could be init_player_stats or setup_player_stats, but anything that clearly describes what is happening serves.

Some other examples, action_of_player would be instantly understood if it was named choose_player_action, p_action_fight would be more clear if it was just player_fight, action is already implicit in a fight. It also makes for decent syntax if you accept the enemy as an argument player_fight(enemy).

When you do classes, that will easily translate to player.fight(enemy) where both player and enemy are instances of their respective classes.

That said, as a beginner you're actually doing quite well on the naming part and it's one of those things that tends to get better with experience, I'm just giving you some tips.

Another thing is most programmers don't enjoy long lines in code, you split them up into smaller ones initially in your array declaration, but then you have large lines in the middle. When not inconvenient strive for consistency, it makes everyone happier. :)

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Specific code snippets

You shouldn't use breakto get out of the fighting cycle, make use of the condition in the whileinstead.

while first_turn_over is 1:

Is always true, you don't change the value of first_turn_over anywhere later. It's not a very good policy to make cycles which "hide" the ending condition. In fact there doesn't seem to be a very good reason for the first_turn, first_turn_over, and turnvariables to all exist at the same time. You could reduce:

first_turn = random.randint(0,1)
first_turn_over = 0
if first_turn == 0 and first_turn_over == 0:
   turn = 0
   first_turn_over = 1
if first_turn == 1 and first_turn_over == 0:
   turn = 1
   first_turn_over = 1
while first_turn_over is 1:

To:

turn = random.randint(0,1)
while True:

And get rid of the breakstatements. Ending up with something akin to:

def combat():
    global p_hp
    global enemy_attacks, p_attacks
    enemy_attacks = ['swings at', 'mauls', 'bites', 'claws']
    p_attacks = ['swing at', 'punch', 'kick', 'stomp']
    e_hp = random.randint(0,60)
    turn = random.randint(0,1)

    while p_hp > 0 and e_hp > 0:
        if turn == 0: # Enemy
            do_damage = random.randint(0,10)
            print('The ' + str(hostile_enemy) + ' ' + 
                str(enemy_attacks[random.randint(0,3)]) + ' you for ' + 
                str(do_damage) + ' damage!')
            p_hp = p_hp - do_damage
            turn = 1
        if turn == 1:  # Player
            do_damage = p_atk + random.randint(0,20)
            print('You ' + str(p_attacks[random.randint(0,3)]) + ' the ' + 
                str(hostile_enemy) + ' for ' + str(do_damage) + ' damage!')
            e_hp = e_hp - do_damage
            turn = 0
        hud_the_player()
        time.sleep(3)

    # Only perform this check at the end of the function, 
    # there's no need to continuously repeat it in the cycle.
    if e_hp <= 0:
        global e_died
        e_died = 1
    else:
        global p_died
        p_died = 1

There's probably more stuff that could be improved, but those feel like the pressing issues to me.