Your WRT54G is a router/switch/access point combined into one. So to answer your question Yes your router is also a access point. If the routers wireless range is not enough for your network area then you can add a WAP54G access point in repeater mode to extend your network range.
A WRT is a router with a wireless access point. This access point only operates in "root mode" which means clients can connect wireless to the access point in the WRT but the WRT is not able to connect wireless to other access points. The upstream link on the WRT always goes through the internet port.
The WAP is a versatile wireless access point. It can operate as standard base wireless access point like the WRT. But it can also operate in a wireless distribution system (WDS) as repeater or wireless bridge. This means the WAP is able to connect wireless to another access point and bridges the wireless signal to its ethernet port or repeats the signals for other wireless clients.
The WRE is a dedicated repeater. Nothing else.
The WET is a dedicated wireless bridge. It connects wired computers wireless to an existing wireless network. Nothing else.
Thus, the question whether to get a WRE or WAP is what you need at the moment and what you think you'll need in the future. Let's say you need a repeater to extend the wireless coverage. You'll get a WRE for that. After a while you realize that the repeater deteriorates the performance of your wireless LAN too much and the throughput through the repeater is too low. Thus you put in a ethernet cable to bridge the distance instead of the wireless link. The WRE is now obsolete. If you have got a WAP you would simply connect the WAP to the ethernet cable and reconfigure it as normal wireless access point to create a single roaming network...