I'd say that 1/16'' tolerance, or 1.5mm respectively, is actually pretty bad. You should rather aim for 0.1mm tolerance, not 1.5mm. The final piece should have no visible gaps.
How to do that?
First of all, of course diligence, proper tools, proper technique, and practice (well, duh). But second, there's a few tricks you can apply, too.
As a beginner, use softwood (pine, for example). Do not make your life harder than it needs to be while you are still learning. Softwood is not only a lot easier to work with and more tool-edge-friendly, it is also a lot more forgiving to little mistakes that you may make, and will make.
When in doubt, you can always make a softwood tail slightly rather too large than too small, and fit it in with a little persuasive force (hammer). That's not the correct way of doing it, but up to a surprisingly large limit, it will work. You can't really do that with hardwood.
If you have visible gaps, even rather huge ones, you can almost always fix the problem with a clamp. You can also do that with hardwood, but the tolerances are much, much smaller.
The proper tools
You need a sharp pencil and a mortise with a precise, sharp edge in the right size.
Which kind of saw to use is somewhat of a matter of taste, but I daresay Japanese saws are superior to western ones. They are usually much, much thinner (like 0.5mm) and have finer teeth, and they cut on pull, which lets you control the saw a lot better. There's nothing like a Dozuki for cutting dovetails.
A jig will help you big time. Commercial jigs like this one which not only guide the saw but also hold it with a magnet give you perfect precision even on your first ever dovetail. However, they make cutting dovetails almost too easy, and they do cost a bit of money. But even a simple small block of hardwood somehow clamped onto the work piece will help big deal doing a perfect perpendicular cut along a defined line.
The proper technique
First, you want to do one half of the joint, do not touch the other piece of wood yet. It is simply not possible to produce something that will fit if you mark the second piece before having finished the first. Decide whether you want to do tails or pins first- there is no right or wrong. Half of the woodworkers will tell you to do pins first, the other half will tell you to do the tails first. Do whichever you feel most comfortable with.
Use a marking gauge to mark the length of your pins/tails so you have the same length everywhere. Copy the required thickness from the board, do not measure. Some gauges have a micrometer screw that lets you adjust the setting. Those are normally not very useful, but here they can be really handy: You can optionally make the tails/pins a tiny bit longer if you want by giving the micrometer screw one or two turns.
The rationale behind that is that if your tails/pins are a tiny amount too short, the joint doesn't fit properly, it looks really bad. Which sucks because there is nothing you can do about it except spend 4 hours on the sander thinning the whole board down. On the other hand, if they are a tiny amount too long, it takes two strokes with the smoothing plane, or 15 seconds with sand paper, and it looks absolutely perfect. Preferably, of course, tails/pins should be exactly as long as they should be, but very slightly too long is harmless whereas slightly too short is a disaster.
Mark the desired shape of your tails/pins. You are in principle free to choose any angle, and any regular or irregular layout or proportions that you want, there are very few "rules" that are the kind you should follow. There are entire religions about which is the correct formula for the width of tails, but I've done them in many variations including irregular widths, and you can't really tell that one or the other is truly "wrong".
As a rule of thumb, try to get a somewhat even distribution for the beginning and do the outermost pins half-width (looks pleasing), and go for somewhat steeper (1:6 or even 1:4) angles if you use soft wood or very thin boards, and rather not-so-steep angles on hardwood. You can use a selfmade jig or a commercial one to get the correct angle, or simply measure. All that is just a rule of thumb, you can in principle do whatever you want.
Although at this point, you do not yet need to be super precise on marking, try to be precise anyway, use a sharp pencil or a kogatana. It never hurts to work with diligence, and you should get the habit of doing precise marks, so do it anyway.
All accessible lines (that is, any lines that have contact with an outer surface) are cut with the saw, then the rest is worked with the chisel. First mark the line with one exact hit, cutting fibers. Then remove the cut-off chip. Begin exactly orthogonal to the board by using a small piece of wood as a guide. After the first millimeter, you are allowed -- and encouraged -- to "cut behind" a little (I am not sure about the correct term to use in English, the German word is "hinterstechen"). Doing so will make it a great deal easier fitting pieces together later while preserving a perfect visible edge on the outside.
TODO: add a sketch (coming on weekend)
Once you have finished one side, hold it against the other side, and carefully, diligently, mark its contour onto the other piece with your sharp pencil. Here, it really counts, so take your time, be precise.
Now do the second piece, but note that the pencil marks you made by drawing around the first piece are of course outside the form of the first piece. Which means that whatever cuts you make, the pencil mark must still be visible afterwards. If the marks are no longer visible afterwards, you will have a visible gap.
When in doubt, stay rather on the safe side. It's better to have the mark visible and the tiniest amount of overshoot than to be too small. With a broad and well-sharpened chisel in your hand, you can carve off very, very minute sheets (less a tenth of a millimeter) if need be. You can always take away, but it's hard to add material that you've cut off. It's possible, only just not nearly as easy.
When trying to fit the pieces together and they don't really like fitting, you will need to remove something somewhere. But where?
Apply a little more force, and then pull the pieces apart. Now look at the little surfaces. They will all be rather uniform in look and texture, except one or two of them which will have a mirror-smooth surface. This is where you tried to force the non-fitting pieces together and they rubbed against each other. Take off a minute slice of wood with a broad, sharp chisel from these locations. Try again, repeat. You won't need to try more often than a couple of times.
The just-over-half trick
When chiselling out, do not start at the beginning of the to-be-removed piece. Instead, start somewhere in the middle, and go down in an angle. Once you are certain that you are done just over half, stop working. Do the remaining pieces on the same side, and always stop once you are just over half done.
Now turn the board around and start from the other side. This guarantees that both sides have a clean, perfect edge, and no visible tearout is possible, no matter what. If you are a bit in a hurry, you can give it a hard hit with the chisel once you're 1/3 done from the second side, that will do away with the to-be-removed piece.
TODO: also add a sketch here
The clamp trick
Clamps can be used not only to press the joint together in the direction of the two boards. A third clamp can be used to slightly compress the dovetails. It will only be maybe half a millimeter over the whole board, but that is enough to hide the visible gaps of a normal manufacturing precision and turn a "meh" joint into a "woah" joint. The glue will hold everything in shape once it is cured.
The water trick
This is a trick that a master carpenter told me (I haven't tried it myself, but once you think about it, it's so darn obvious that it simply must work). Wood expands when exposed to moisture. So, if you have visible gaps, give it a mild stroke or two with the sponge, and wait a few minutes.
Obviously, having a glue that isn't too sensitive to water (or one that even prefers the presence of water, like PU glue) is necessary here.
The dirty trick
If, after all, you still have "meh" quality joint with visible gaps, fill the gaps with a mixture of sawdust and glue. The sawdust will give the glue a convincing color, which will make the gap look a lot less prominent.
For that reason, it's always a good idea to preserve some of the sawdust or wood shavings.