How do I get dragon type pokemon's?

Vibrava and Flygon: Trapinch evolves into Vibrava at level 35 and Vibrava evolves into Flygon at level 45.
Altaria: Evolve Swablu at level 32 or capture a wild Altaria on the fifth level of the Sky Pillar.
Kingdra: Trade Seadra with a Dragon Scale attached.
Bagon, Shelgon, and Salamence: Bagon is in the chamber of Meteor Falls in which you find TM02 (Dragon Claw). It evolves at level 30 into Shelgon, which evolves into Salamence at level 50.
Latias: In Sapphire it is found on Routes 110, 111, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, or 134 after defeating the League and watching television downstairs at your residence in Littleroot.
Latios: Same as above in Ruby.
Rayquaza: Top level of the Sky Pillar.

Use super rod in Sotopolis:Gayrados.
In the Meteore Falls you can catch bagon train it too level 50: Salamence.
Catch horsea and train him and give him water stones:kingdra.
Catch Trapinch and train it to level 42:Flygon.
This is all i can remember.

Toptrainer

Just find a trapinch and bagon train those two until they evolve to their third forms and you now have dragon pokemons.

There are several dragon types available:
Rayquaza - only one - at the top of Sky Pillar after beating elite 4 (its at level 70 and knows rest, extremespeed, fly, and outrage) - hard to catch!
Salamence - catch a Bagon from the room in Meteor Falls where you found TM02; it'll evolve to Shellgon and then to Salamence.
Latias/Latios - only one; running around wild in Hoenn after beating elite 4 - use master ball to catch b/c it'll run away from you as soon as you find it!
Kingdra - find a dragon scale (attached to wild Bagon) and trade a seadra holding the scale - viola!
Flygon - find a wild trapinch - it'll evolve into Vibrava and then Flygon.
I may have forgotten some - but that'll get you started!

To answer your 3rd question no your mom will not battle you nor will team rocket want you to join ,BYE!

I will list some Dragon Pokemon that you can get:
Rayquaza
Altaria
Vibrava
Flygon
Latios
Latias

Rayquaza- After defeating the Elite Four, Fly to Pacifidlog Town, then head northeast. You will find and entrance amongst all the boulders in the water; beyond lies Sky Pillar. Be sure to have your Mach Bike handy bfore you go.

Altaria: Evolve Swablu at level 32. Or, check within the upper levels of Sky Pillar.

Vibrava and Flygon: Evolve Trapinch at level 35 to get Vibrava, which from there you should level up to level 45 to recieve Flygon.

Latios and Latias: Latios and Latias are found anywhere in Hoenn after you defeat the Elite Four. Latias is found in Sapphire Version, and Latios is found in Ruby Version.

Evolve Trapinch, Swablu, Seadra(Seadra By Trading It Whilst Holding A Dragon Scale Or Fang(Cant Remember), Bagon, Dratini.

Catch Bagon, Latias, Latios, Dratini, Rayquazza.