I'm not sure whether you're concerned here about pixel count, resolution, or file size.
Pixel count is the number of pixels vertically and horizontally, and is the best measure of the actual data size of the scan. For example, a scan of a 35mm negative made with a 2000 ppi scanner will be roughly 2000 pixels high x 3000 pixels wide. This number is determined by the settings used when the film was scanned, and it shouldn't vary subsequently unless you "resample" the scan to change its pixel count.
To find out an image's pixel count, you need an application that can read it out. Sometimes your operating system can show it directly as part of the file's property information; Photoshop and most other image editors can doisplay it; I don't know anything about Picasa, since it's Windows-only, so I don't know where to tell you to look for this info.
"Resolution" is pixel count applied to a specific output size. To choose an easy-math example, if you took the above scan and printed it on 10x15-inch paper, the resolution would be 200 pixels per inch (2000 pixels divided by 10 inches, and 3000 pixels divided by 15 inches.) You can work this out for any output size if you know the pixel resolution, as above. So, a scan of a given pixel size will have different resolutions for different output sizes.
File size is the number of kilobytes or megabytes that your scan occupies on a disc. This is what I suspect you're seeing when you say the image varies from 1.5mb to 6mb. This is not necessarily any thing to worry about -- it's quite common for scans with the same pixel dimensions to have different file sizes.
The reason is that most digital-image file formats, such as JPEG, use mathematical compression techniques to make the file size smaller. An image that has large blocks of similarly-colored areas will compress much more than one that has many areas of tiny, intricate detail. So, for example, if your first picture shows mostly blue sky and the next shows a crowd of people's faces, the first picture will compress to a much smaller size than the second one, even if the scans have exactly the same pixel count.
So, to find out what's going on, the first thing you need to do is examine the images you get back on the disc and see whether or not they all have the same pixel count. If they're being scanned at different pixel counts, something really weird is going on in the lab and you're entitled to find out what it is. It actually would be easier for them to scan everything at the same pixel count, so if you're getting different counts, it's probably a malfunction or a mistake.
Once you know that you're comparing images of the same pixel count, you can look at the file sizes and see if they're realistically varying because of compression. If you've got two files with the same pixel size, and one of them has a file size of 200kb and prints poorly, while the next has a file size of 6mb and prints well, it suggests that your lab is using compression settings that are much too high for some of your images. This would degrade image quality and might account for your poor results. What you can tell the lab to make them stop doing that, though... well, that's an entirely different problem!