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I am looking for an expression (maybe a 成语) which can describe the idea of, "I know I have a long way to go, but if I keep trying, I can succeed."
For example, "我的发音还差得远，可是" Meaning, "My pronunciation has a long way to go, but I will keep working hard at it."
I have found this one expression that seems to express it, is it appropriate?
只要功夫深，铁杵磨成针
Or perhaps:
磨杵作针
Please let me know how I can best express this idea in a way that the average speaker can understand.
In addition to @songyuanyao's extensive list, there's a surprisingly large number of sayings that are specifically about succeeding by persevering:
In the question's specific context, I would say the first would be the best choice (我的发音还差得远，可是勤能补拙). But 2-5 can substitute for it quite well too.
The last two still mean succeeding with hard work, but they'd normally used after the fact, e.g. as 畢業時，他的發音艱難玉成，十分標準 "By graduation, his pronunciation became excellent after putting in lots of work".
The two you listed (只要功夫深，铁杵磨成针 and 磨杵作针) are also excellent choices; they are both variations of the same underlying phrase so you can choose whichever you find more pleasing. Other forms include:
艱難玉成 for instance I've only seen a couple of times, e.g. in an article on who went studying abroad. The ones that could also be lost on a reader might be 駑馬十駕, 功在不舍 and 跬步千里/跛鱉千里, which are also a bit rare, though they are relatively well known quotes from Chinese classes. For reference I'd consider 勤能補拙, 日起有功, 事在人為, 人一己百 and all the longer ones to be well known.
– Semaphore
Jul 15 '14 at 3:25
只要功夫深，铁杵磨成针 is good, and also:
坚持不懈
锲而不舍
水滴石穿
百炼成钢
熟能生巧
精卫填海
卧薪尝胆
有志竟成
卧薪尝胆 came from a story about revenge, but as a 成语, it can be used for the meaning of keep trying hard to achieve one's goal. Such as: 中国足球要想冲出亚洲，必须卧薪尝胆，刻苦训练。
– user4072
Jul 15 '14 at 1:58
I think most Chinese people would express this with a simple "慢慢来," which pretty much includes all the elements you're looking for ("I know I have a long way to go, but if I keep trying, I can succeed.").
If you want something more idiomatic, consider:
一口吃不成胖子
ABC
1 Nothing can be accomplished in one single effort.
2 Rome wasn't built in a day.
CC-CEDICT
1 lit. you cannot get fat with only one mouthful (proverb)
2 fig. learn to walk before you run
It's not as direct as you might like, seeing as the meaning is more like "nothing can be accomplished that quickly, but it will work if you like it.
Akin to this we also have
冰冻三尺，非一日之寒
CC-CEDICT
three feet of ice does not form in a single day (idiom); Rome wasn't built in a day
冰冻三尺，非一日之寒 is more typically used for calamities or other negative events. For instance, to describe the Fall of Rome (羅馬的顛覆不能歸咎於阿提拉：羅馬的衰弱乃冰凍三尺，非一日之寒 "The Fall of Rome can't be blamed on Attila the Hun: Rome's decline was a long time brewing"). A common usage is to alleviate blame from a single event or person.
– Semaphore
Jul 14 '14 at 2:25
Yugong moves mountains
愚公移山
Yúgōngyíshān
I'm surprised this didn't come up earlier, but 愚公移山 (Baidu Baiki; Wikipedia) is one of the most famous 成语 and it's about perseverance. It tells the story of Yugong who moved mountains piece by piece, and succeeds because he doesn't give up. There are a lot of variants of the story. Wikipedia describes is like this:
The myth concerns a Foolish Old Man of 90 years who lived near a pair of mountains (given in some tellings as the Taihang and the Wangwu mountains, in Yu Province). He was annoyed by the obstruction caused by the mountains and sought to dig through them with hoes and baskets. When questioned as to the seemingly impossible nature of his task, the Foolish Old Man replied that while he may not finish this task in his lifetime, through the hard work of himself, his children, and their children, and so on through the many generations, some day the mountains would be removed if he persevered. The gods in Heaven, impressed with his hard work and perseverance, ordered the mountains separated.
There much learning materials available about this 成语, such as a ChinesePod episode, various YouTube videos (e.g. Gingle ALE and 听故事学中文), Chinese at Ease, etc.
So maybe we can complete the sentence like this:
我的发音还差得远，可是像愚公移山一样，只要坚持下去，就迟早会获得成功。
My pronunciation [is] poor, but like Yugong moving the mountains, if only [I] persevere, sooner or later [I will] attain success.
千里之行始于足下 fits your contexts well, although the original meaning doesn't really mean what it means today.
白居易's reference represents the modern meaning better and it fits your context.
《温尧卿等授官赐绯充沧景江陵判官制》：“夫千里之行，始於足下。苟自强不息，亦何远而不届哉？”
Translation mine:
(As we know) A journey of a thousand miles starts from underneath of the feet. If you self improve relentlessly, then how is it possible that you can not reach the faraway (goal)?
