Do you have to add the "enchantment" cost to the weapon cost? Does the player just pay the amount? Do you have to buy a completely new weapon? For example for a +1 magic weapon, it costs 360... Is that the total cost for all weapons?
There are two systems for items in 4e. The original system and the new system.
In the original system there were two ways to acquire magic items. You either found them while adventuring or you bought them from a shop. The rules on what was available from a shop were pretty thin and left completely up to the DM's discretion. The limiting factor here was that you could only use a Daily item power once per milestone (2 encounters) completed.
However, when Essentials came out, there was a change. They introduced item rarity and all magic items now have a rarity (common, uncommon and rare) and they removed the restrictions on daily item powers. You can get common items the following ways:
Find a shopkeeper who stocks the item (DM discretion here), and purchase it for the listed gold cost (this is the cost listed on the item, the mundane weapon components are factored in here).
Enchant the item yourself using the "Enchant Magic Items" ritual or similar. For this you need a caster able to cast the spell at the level you need for the item, the item you'd like to enchant, and enough gold to cover the full cost of the enchantment (the item's cost).
Find them while adventuring.
Rare and Uncommon items can only be found while adventuring (though some DMs will make an exception and allow you to purchase Uncommon items with strings attached. Our DM uses a waiting period for current or near current level uncommon items).
To answer your question explicitly, yes, the players just pay the amount listed on the item (for common items). However, this is an opportunity to bust out the roleplaying and have them find a shop, if you want they can haggle with the merchant (and maybe get a better deal! (just not too good)).
As far as the cost, this is dictated not by the enchantment level (the +1), but by the item's level, so all L1 items are 360 gold, all level 2 items are 520 gp.
Enhanced weapon/armour's cost includes the cost of the base item, unlike previous edition. So a +5 longsword would cost the same as +5 resonating dagger and would cost the same as a +5 Legion Plate armor, namely 225k gp.
The cost to get a new enchantment depends on what you have and what you want to do.
Enchant a non-magical item
If you have a non-magical base item you can enhance it using the Enchant Magic Item level 4 ritual. This ritual cost exactly the same as buying a new one, though, so meta game wise it is better to buy a new one since you get to keep your old item.
Upgrade a magical item
The Enchant Magic Item ritual can also be used to upgrade an existing magical item, e.g. upgrade a +2 Flame Tongue Longsword to +3, and you only pay the difference in cost. But it cannot change or upgrade the type of enhancement as written, for example +3 Flame Tongue Longsword cannot be upgraded to +4 Sunblade Longsword by the rules.
Transfer enchantments from/to another item
You can use the Transfer Enchantment level 4 ritual to transfer an enchantment from one magical item to another mundane or lower level item, as long as the result is valid. For example if you have a +1 Longsword and a normal Spiked Chain , you can transfer the "Magic Weapon +1" enchantment from the longsword to the spiked chain. The longsword becomes mundane, and the spiked chain becomes +1. Component cost of the ritual is only 25gp.