Getting
HM5 in Bellsprout Tower
Just North of Violet City, there is a Sprout Tower with trainers
using only Bellsprouts (#69). Battle your way through until you reached the top
level. There are lots of Itemballs in the tower and they contain Parlyz Heal, X
Defend, Potion, and Escape Rope. You will have a little encounter with your
rival but there will not be a battle. However, if you want your HM5, you
will need to beat the Elder of the Sprout Tower (Sage LI) at this level.
The Elder will use two LV 7 Bellsprouts and a LV10 HootHoot
(163).

Getting
your first badge - ZephyrBadge
When you think you are ready, you can battle the gym in this city
to get your first badge. All the trainers and the gym leader use Bird type
Pokémon. After you defeat Falkner (uses LV7 Pidgey and LV9 Pidgeotto), he
will hand over the ZephyrBadge. It will increase the attack of all your
Pokémon and allows you to use Flash in and out of battles. You will also
be given a TM31 (Mud Slap) which is an attack that lowers the accuracy of your
opponent while it causes some damage (much better than Sand-Attack).

Getting
your Egg back
The minute you leave the Violet Gym, you will get a call from
Prof. Elm. Make sure you have space for another Pokémon, then talk to his
assistant at the Pokémon Center. Answer "Yes" to his question, and you
will be given your Egg back. Prof. Elm wants you to carry this Egg with you in
your journey because it is the only way it will hatch. This Egg will soon hatch
into Togepi.
Afterwards, you may try to head West to Route 36, but you will be
blocked by a funny shaking tree. This is actually a Pokémon, but you cannot
battle it yet. At this point, head South through the North Entrance to the
Ruins of Alph by Route 32