That's not necessarily indicative of it being a competitor, you'll get plenty of enquiries like this from all sorts of people. Some people won't want to make an appointment because they are just in the early stages, or are worried they may be pressured into something - many genuine customers simply have no idea of the costs involved in getting a site up there, and just want to get a rough idea before they dive into choosing a vendor.
There's plenty of different 'buying processes' out there, each prospect is different, you just have to adapt to each customer accordingly. Just make sure whatever you are doing fits well with the way you wish to conduct your own sales processes, but be prepared to make some compromises when required.
Do you talk to them over the phone, or do you see them in person? I really have to change the way to I do things in order to progress.
There's no set answer to this, and in the end, it's something you have to make a judgement call on, which is where experience will come into play. I always try to engage the prospect over the phone or email for as long as I need to suss them out and work out what stage of the buying process they are (10-15 minutes, or a couple of emails usually). I also try to assess whether we'd be a good 'fit' and whether or not I can actually do what it is they want, at a price they'll be happy to pay. If they tick all the boxes, we can spend more time talking, phone, face to face, whatever seems suitable.
I'm not a fan of driving around all day in meetings with potential clients, it's very time consuming, so I save that for only when I am confident I have a good sale on the cards. Of course, if a big company rings and wants a meeting, I'm in my suit and straight down there. You just have to take each prospect individually.
On the flip side, some developers seem to do everything by email or instant messaging. Again, different clients, different methods.