You'll need to build weapons to obtain them. You'll have to purchase the Blueprint on the Market (Market > Weapons > Primary/Secondary/Melee Blueprints), through Clan Research or find specific Blueprints/Components in special missions. You can also directly buy a pre-built weapon with Platinum. For more info on building weapons, see the Scientist trophy.

In order to level up a weapon you'll have to keep using it. The more kills you get with a weapon, the more Affinity (experience) will go towards the weapon. It's advised to use your Warframe damage abilities as little as possible. If you use your Warframe damage abilities, all possible Affinity you could've obtained from killed enemies will go towards leveling up your Warframe and not your weapon. If you play solo, use your weapon as much as possible.

If you don't like your weapon and want to use it as little as possible, you can also choose to play in a group. Every kill made by your team mates, no matter how the enemy died, the total Affinity will be devided evenly over your Warframe and weapons (25% each).

Make sure to stay close to your squad members, else you won't receive Affinity from their kills. You might also want to remove your Sentinel weapon and certain mods, since they can steal kills, thus stealing Affinity from you.

You obviously want to play a mission with many enemies. Survival and Endless Defense missions are a good choice since they spawn enemies infinitely. A somewhat quicker method, even though it doesn't spawn enemies infinitely, are Mobile Defense missions. Keep repeating these over and over and keep using your weapons to kill enemies as much as possible. You will probably want to play in a squad for the reasons mentioned before, but also because playing missions with more squad members will increase the number of enemies.

Some recommended levels by the community to quickly level your weapons are:

Baal, Europa (Mobile Defense)
Kappa, Sedna (Mobile Defense)
Cyath, Eris (Mobile Defense)
Kiste, Ceres (Mobile Defense)
Palus, Pluto (Survival)
You can also play Dark Sectors. These are the levels controlled by clans. These missions will offer bonus experience for kills and certain weapon types. The bonus percentage differs per sector, and so does the weapon bonus, so make sure to pick the one that suits your weapon. Also try to play Survival or Mobile Defense missions.

Another quick way to level up are Orokin Derelict Defense (ODD) missions. These however are more suited for the advanced players. You can always ask around and see if others are willing to invite you to one and level your weapons.

You can also buy Affinity boosters in the Market > Equipment > Boosters. These will cost you Platinum though.

Be sure to mod your weapons whenever possible to increase their efficiency. Another great trick -depending on the situation- is to use stealth with melee weapons. This will highly increase your melee damage which makes getting kills easier, thus quicker and possibly more experience for your weapon. Stealth kills are performed on unalerted enemies, which can be quite tricky. This can be made easier with the Shade Sentinel, Loki's Invisibility ability and Ash's Smoke Screen ability.

The total Affinity required to level your weapon to Rank 30 is 450,000. You can check the total amount at your Profile > Primary/Secondary/Melee Weapons.

Note: You only start out with 5 weapon slots (Arsenal > Inventory > Weapons) while you need to build and level at least 10 weapons. You can purchase extra slots with the Platinum. You can also delete any unwanted weapons to make room. Do not delete weapons of a certain weapon type if you haven't leveled it up to 30 though.