Here's my little tour around a few places I visited on my last trip to Bangkok. I was staying on Soi 4/Nana and my hotel (Heaven@4) was only 5 minutes walk to Nana BTS.
I've only visited Bangkok twice, and only for a few days each time. The first trip was mainly occupied by the nightlife and just being in awe of how large an area Sukhumvit covers (I'm used to Patong :P), and also how crazy busy it is! So I can't speak for the rest of Bangkok, but I do like this area as a tourist. It's close to the Skytrain (BTS) and Asok has an interchange to the underground (MRT) - so good reach for transport. Asok also has the Terminal 21 shopping mall - though it has a lot of high end shops so can be expensive.
My second trip was with a Thai friend who showed me most of the tourist hot-spots, and after visiting each one I could tell you could be easily ripped off by tour guides/taxis if you didn't know your away around - but it's really not that hard to get around yourself! I think it's just being in a new city for a lot of people - it can throw your bearings off. Once you know which direction trains are going and where everything is relative to your location it gets easier.
I like this map for showing the MRT/BTS and Airport Link:
Since I was only there for the weekend (and slightly hungover some days) I did all this over the course of two days (but could do it in one). The first thing I wanted to see was the King's Palace (Grand Palace). I talked with my friend and the suggested route was via the MRT and then get a Tuk-Tuk, since the Palace isn't really near any train stops. So from Nana I got on the BTS to Asok (15 Baht, though you could probably walk it in 10 minutes), and then swapped to the MRT underground and got on the southbound train to Hua Lamphong (27 Baht). From there we just jumped in a Tuk-Tuk and prayed for our lives as it dashed between traffic (I'd recommend a Tuk-Tuk just for its nimbleness, as traffic is always hectic). I think it was around 100 Baht but that was with a Thai bargaining - I'm sure he was hoping for Farang prices. It took around 10-15 minutes with the traffic, but we were soon outside the Palace and oh my god.. it's HUGE.
The Palace closes at 3:30pm so make sure you don't arrive too late. When you enter you'll be forced to borrow (for free) any clothing that may be required to cover the usual body parts for this type of place (so no shorts, short skirts, and shoulders must be covered etc). After that you make your way to the entrance and get whacked for 500 Baht per person (I thought that was a bit steep but it is pretty big), although Thais get in for free (as with most places). Inside it's just walking the grounds and checking out the fabulous buildings - lots and lots and lots of buildings, mostly gold. In the middle there is the Emerald Buddha temple (Wat Phra Kaew). My Thai friend was saying this is one of the most important and sacred temples because of the emerald Buddha and its significance to Thailand as a country (i.e. don't be taking Buddha out of the country!). You also can't take pictures, yet I was amused to watch a tourist try and snap a picture with his phone only for the guards to run over and confiscate it. It didn't look like he was getting it back at one point, but he seemed to convince them he wasn't taking anything (or he deleted it for them). Elsewhere in the grounds there are a few museums with ancient weapons (swords, spears, etc) if you're into that sort of thing, otherwise just more buildings...
Next up (if you did the King's palace early) would be Wat Pho, which is south of the Palace and a good 15-20 minute walk. If you go west around the Palace and then south along the river, you'll see small market along the street selling the usual touristy things and old junk that no one ever seems to buy. At the end of the street, cross over and you'll be at Wat Pho.
I actually liked Wat Pho more than the Palace - it just had more unique things to see. I was also very lucky to be there at a time where there was a Buddhist ceremony for raising money for the poor. My Thai friend was very excited and she was really happy she caught it. Apparently only happens once per year but is a pretty big deal for them. After the usual chanting the monks were spraying water everywhere (yes I haven't got much clue about Buddhist ceremonies, but it was obviously a blessing and everyone was trying to get in on the action). I do like this sort of thing though - I just really admire how deeply rooted religion is in their culture, and I'd love to learn more about it.
After Wat Pho I would then visit Wat Arun, which seems to be a bit more famous. Wat Arun is easy to get to from Wat Pho, as the street opposite which comes down from the Palace also hosts a boat pier, so a short ferry crossing (it wasn't expensive) and you'll arrive literally at the door of Wat Arun. It's a little different in setup than Wat Pho - mainly focusing around the large "pyramid" in the middle which you can climb (*you* can climb... I was knackered). Still, lots of interesting statues to look at, though not much different to every other temple. You can hire traditional clothes there and have your picture taken against the backdrop which I thought was a nice touch. The guy I saw partaking looked a bit of a tool dressed up like that though - I passed.
Both Wat Arun and Wat Pho cost around 100-150 Baht each if I recall.
After getting the ferry back to the east side of the river we then got the Chao Phraya Express Boat which goes south of the river and stops at each pier along the way (35 Baht if I recall). If you got on near Wat Arun/Wat Pho, which I think is Tha Tian, you want to get off a good 20 minutes later at Central Pier which is next to Saphan Taksin BTS, and then you're back to familiar ground. Saphan Taksin to Siam BTS is about 34 Baht, and from there you could either have a look around the big Paragon mall (also has a nice Ocean World attraction) or just head back to Nana/Asok (25/28 Baht). You could also visit MBK which is one stop from Siam (or walk it) for any gadget needs (it's a huge electronics place).
I'm sure there's plenty more to see elsewhere in Bangkok, but that's for next time.