When I want to say "working out", "exercising", or "training" in Chinese, I come across these two words:
What are the similarities and differences between these two words and their English counterparts?
运动 means "to exercise" or "to work out" . It is basically a method of strengthening one's muscles through body movements, to gain power and or improve endurance in sport. It can also just for keeping one's body healthy and fit.
锻炼 means "to train" or "to practice" . It refers to "improve skill and ability through training and practicing" It can apply to any thing that require skill and physical ability. For example, you can 锻炼 your language skill, chess playing skill or your eye sight. 运动 is one of a methods for someone to 锻炼 his physical ability.
运动 as a verb can be used without object. e.g. "you need to exercise more" (你需要多运动)
锻炼 as a verb usually be used with object. e.g. "you need to train your hand eye coordination more (你需要多锻炼你的手眼协调)
I think Tuttle Learner's does a good job explaining these two terms:
Let's start with:
[comp: 运 move + 动 move]
do physical exercises
Nǐ jīngcháng yùndòng ma?
Do you exercise often?
and then we have:
[comp: 锻 shape metal + 炼 smelt]
undergo physical training, do physical exercises
Nǐ yào chéngwéi yì míng hǎo yùndòngyuán, jiù děi tiāntiān duànliàn.
If you want to become a good athlete, you have to train every day.
There's also a helpful note in KEY for 锻炼 that says
to temper, to steel, to toughen (oneself) up
An easy way to remember is that 运动 can also be the word for movements (political, artistic, social), for instance The African-American civil rights movement is known as the 非裔美国人民权运动.
运动, as a noun, usually refers to a certain sport activity, like 乒乓球是一项流行的运动 (Table tennis is a popular sport.), as a verb, it means doing sports, like 我下午去运动 (I'm going to do sports this afternoon.)
锻炼 can also be used as a noun or verb. It usually refers to a certain activity or the action that can improve your ability, no matter it's regarded as a sport or not, like 学生们在体育课上得到锻炼 (The students take exercises on physical education classes.), where the students may simply do something asked by their teacher like stretching their arms, or, 你应该多锻炼你的平衡感 (You should improve your sense of balance.), where the objective to optimize is not a sport at all.
Training would be 训练 in Chinese, which is (at least to me) a series of systematically arranged activities aiming at strengthening or improving an ability.
In your case, I think saying both 我去锻炼 or 我去运动 are correct.