The Rebbe's guidance on how trust in G-d and one's own efforts fit together:
Trust doesn't replace effort. Except for rare tzaddikim who live above worldly concerns, G-d wants us to make a natural "vessel" for His blessing (IK 256); relying on miracles is discouraged (IK 3266). Providing for one's family is itself a mitzvah that calls for both faith and real effort (IK 2746, 2750).
But the effort is only the vessel. Healing and livelihood come from G-d; the doctor and the job are His channels (IK 801, 846, 1071, 1612). Following medical advice is a Torah obligation (IK 9629), and trust goes together with it (IK 9274, 10272). Make every natural effort, but put your trust in G-d's goodness, even through delays (IK 1525).
Don't overdo it. Overexertion in earning a living can actually hinder success; trust brings the blessing sooner (IK 7304). Normal effort is enough — keep your main focus on Torah (IK 922), and a young man learning Torah shouldn't be anxious about his future livelihood (IK 3952).
Spiritual vessels too. Torah study, mitzvos, tzedakah and checking tefillin and mezuzos are also vessels for blessing, alongside the natural ones (IK 1385, 1511, 5544, 9142).
When the natural route looks closed. Even when a shidduch seems impossible by natural means, trust G-d and keep taking practical steps (IK 2051). Don't be discouraged by negative medical predictions (IK 5499). After a financial loss, don't despair — strengthen trust and give tzedakah (IK 5805); financial struggles are tests to be met with faith and action (IK 7267, 9992).
Get good advice. Consult experts and trusted friends (IK 6128, 8236, 10522), and make business decisions by the actual conditions while widening your channels of blessing (IK 6922). An institution shouldn't cut back essential work out of caution alone; with trust and proper effort, it can expand (IK 2090).
See also: Bitachon – What trust in G-d means; Livelihood – Worry about money and trusting in G-d; Health – Following the doctor's advice.