The biggest issue with Nami is that her skills put the skill in skill shots--they are really unforgiving, both in difficulty to land and mana costs. However, getting good with Nami's skills is very rewarding as they are some of the strongest abilities in game when mastered correctly.
Her passive is actually pretty great. Remember to use it to get back to lane faster or to give an initiation some extra movespeed (casting an E on a jungler as he ganks, for example).
Rremember that *any* skill procs it, so your allies get this from standing on a Q and so on. It can be useful to Q an enemy off your ADC and make sure that both the enemy gets snared and your ADC gets a speed boost.
Her Q's hitbox is misleading and small. It's also a skill that's just hard in general to land. Here are some tips to landing it:
Always cast E on yourself before you Q if you have enough mana--or if it's a jungler gank and you can't afford to miss. Auto attacking with E before you land your Q will make your Q much easier to land.
Always aim behind the target in a way that they will have to severely sidestep in order to dodge. Try to make the target move forward. This way, even if your Q misses, you may have made them path in an awkward direction making them more vulnerable.
Try to predict the player's movements and cast Q where they are headed. If an ADC is about to move in for that creep that is at 100 HP, you can put your Q right in their AA range max is, so when they step in to last hit, a bubble is waiting for them. Likewise, when someone is escaping, try to predict where they are running and cast it in their path so they either have to stop or walk around it.
Do not impulse cast. This one is pretty big: never throw out a blind Q. It costs a lot of mana and it has a long CD. Make sure to save it for an opening where the enemy is either out of position, running blindly, and/or may not react quick enough to harass such as CSing.
It is also worth mentioning that in team fights, a good way to always ensure you'll land a Q, is to babysit your ADC and AP mid. Basically 1-2 champions will run at them, and if you put a Q at the target you're trying to save's feet, you can usually stun both.
Here's an example of a max range Q with good positioning. By putting it right on the edge of the target, Ashe has to move FORWARD to escape it, which will make her more vulnerable to being engaged on. If she tries to run backward, she will get snared by the Q. I've also ensured that she's momentarily distracted by CSing and may not even realize she's vulnerable to being caught.
Her W is pretty straight forward, but I see many Nami players not realizing its power as a poke. I try to always cast it as a heal on my target first then have it bounce to an enemy. While this is easy in a team fight, in laning, be sure you are using it! It's always a waste if you just cast W to purely heal.
Remember it has two bounces after the initial cast, so you can get two heals out of it if you position correctly. For example, cast it on your ADC who is near their ADC, watch it bounce to their ADC, then be in range of the ADC to receive a bounce heal back to you.
Also, much like Lulu's Pix and Q, you can use W to snipe people who have gotten away. Sometimes when people are ganked, they get away with ~100 HP and are under tower away from everyone but you. A good W on yourself will often execute them. Risky? Yes, but you'll get used to the sudden burst and range it has. I have saved many failed ganks this way.
Here is an example image of its range. From this range, I can heal myself and also damage Ashe!
Her E is one of the strongest abilities in League of Legends that no one knows about. There are really two ways to use it: put on on your lane partner or put it on yourself.
If you are laning with an extremely long range champion,it's good to put it on yourself. If you are closer to the enemies since you are zoning them, feel free to use it on yourself to set up plays; you should be autoattacking anyway for poke and it will help you set up plays through slowing them for easy Q's.
Since E also goes off who it's on for magic pen, casting it on yourself is better for damage! But you can't avoid how long ranged Caitlyn is for example, so you have to use your best judgment. Ultimately you want the slow applied as much as buy lol power leveling possible.
The core of E is that it lets you manage your Q and R better by slowing the intended targets, so make sure it's being applied by your ADC or you when you are engaging or disengaging.
Do not underestimate this skill. It is useful in all stages of the game. It lets you chase down people and it also lets you escape, so always remember it exists. Putting E on and getting an attack off on their jungler trying to tower dive you can be the difference between life and death--or at least mutually assured destruction.
Her R is her worst spell. It's an amazing ultimate with several fatal flaws: it moves slow, it costs a lot, and its CD is completely devastating. Her R is one of the main reasons Nami isn't ban status.
However, her R is absolutely incredible when it hits multiple targets. In order to hit multiple targets, it has to surprise enemies and/or be confined to narrow spaces since it moves so slow it is easy to dodge. I prefer to cast it either in the heat of battle (when enemies are doing too much to pay attention to dodging it) or in closed spaces (such as Baron or Dragon pits).