A Study of TAK-788 in Adults With Non-Small Cell Lung Cancer

General Inclusion Criteria all cohorts: dose escalation, antidiarrhea prophylaxis, dose escalation combination, expansion, and extension: 1. Have histologically or cytologically confirmed locally advanced (and not a candidate for definitive therapy) or metastatic NSCLC disease (Stage IIIB or IV) or other solid tumors. For all cohorts except Expansion Cohort 7, the locally advanced or metastatic disease is NSCLC. For Expansion Cohort 7, the locally advanced or metastatic disease is any solid tumor other than NSCLC. 2. Must have sufficient tumor tissue available for analysis. 3. Must have measurable disease by response evaluation criteria in solid tumors (RECIST) v1.1. 4. Male or female adult participants (aged 18 years or older, or as defined per local regulations). 5. Eastern Cooperative Oncology Group (ECOG) performance status 0 to 1. 6. Minimum life expectancy of 3 months or more. 7. Adequate organ function at baseline. 8. Normal QT interval on screening electrocardiogram (ECG), defined as QT interval corrected (Fridericia) (QTcF) of less than or equal to (≤ ) 450 millisecond (ms) in males or ≤ 470 ms in females. 9. Willingness and ability to comply with scheduled visits and study procedures. Part 1: Dose Escalation Cohort Specific Inclusion Criteria: 1\. Refractory to standard available therapies. Part 2: Expansion Cohort 1 Specific Inclusion Criteria: 1. Have a documented EGFR in-frame exon 20 insertion by a local test. 2. Previously treated with one or more regimens of systemic therapy for locally advanced or metastatic disease. 3. Prior treatment with an EGFR TKI is allowed unless the participants had an objective response and subsequent progression as assessed by the investigator or treating physician. Expansion Cohort 2 Specific Inclusion Criteria: 1. Have one of the following documented by a local test: 1. A HER2 exon 20 insertion; 2. An activating point mutation in HER2. 2. Previously treated with one or more regimens of systemic therapy for locally advanced or metastatic disease. 3. With an EGFR exon 20 insertion: Prior treatment with a pan-HER TKI (example, afatinib, neratinib, or dacomitinib) is allowed unless the participants had an objective response and subsequent progression as assessed by the investigator or treating physician. Part 2: Expansion Cohort 3 Specific Inclusion Criteria: 1. Have one of the following documented by a local test: 1. An EGFR exon 20 insertion; 2. A HER2 exon 20 insertion; 3. An activating point mutation in HER2. 2. Previously treated with one or more regimen of systemic therapy for locally advanced or metastatic disease. 3. For participants with an EGFR exon 20 insertion: prior treatment with an EGFR TKI is allowed unless the participants had an objective response and subsequent progression as assessed by the investigator or treating physician. 4. For participants with a HER2 exon 20 insertion or HER2 activating point mutation: prior treatment with a pan-HER TKI (example, afatinib, neratinib, or dacomitinib) is allowed unless the participants had an objective response and subsequent progression as assessed by the investigator or treating physician during treatment with that prior TKI. 5. Have either previously untreated intracranial CNS metastases or previously treated intracranial CNS metastases with radiologically documented new or progressing CNS lesions. 6. Have at least one target (that is, measurable) intracranial CNS lesion (greater than or equal to \[ ≥ \]10 millimeter \[mm\] in longest diameter by contrast enhanced magnetic resonance imaging \[MRI\]). Part 2: Expansion Cohort 4 Specific Inclusion Criteria: 1. Have one of the following documented by a local test: an activating mutation in EGFR including exon 19 deletions or exon 21 L858R substitution (with or without T790M), or an uncommon activating mutation other than exon 20 insertion including, but not limited to, G719X (where X is any other amino acid), S768I, L861Q, or L861R. 2. Treatment naive for locally advanced or metastatic disease or previously treated with one or more regimens of systemic therapy for locally advanced or metastatic disease. Part 2: Expansion Cohort 5 Specific Inclusion Criteria: NSCLC participants with EGFR exon 20 activating insertions, who have previously shown an objective response to an EGFR TKI and subsequently progressed, without active CNS metastases. 1. Have a documented EGFR in-frame exon 20 insertion by a local test. 2. Previously treated with one or more regimens of systemic therapy for locally advanced or metastatic disease. 3. Previously showed an objective response to an EGFR TKI, and subsequently progressed as assessed by the investigator or treating physician. Part 2: Expansion Cohort 6 Specific Inclusion Criteria: NSCLC participants with EGFR exon 20 activating insertions, who have not received prior systemic anticancer treatment for locally advanced or metastatic disease, without active CNS metastases. 1. Have a documented EGFR in-frame exon 20 insertion by a local test. 2. No prior systemic treatment for locally advanced or metastatic disease. Part 2: Expansion Cohort 7 Specific Inclusion Criteria: Participants with solid tumors other than NSCLC with EGFR/HER2 mutations against which TAK-788 is active, without active CNS metastases. 1. Have a solid tumor that is not NSCLC, including, but not limited to, bladder/urinary tract cancer, breast cancer, gastric/esophageal cancer, biliary tract cancer, and head and neck cancer. 2. Is refractory to standard therapy. 3. Have EGFR or HER2 mutations, documented by a local test. Part 3: Extension Cohort Specific Inclusion Criteria: 1. Have a documented EGFR in-frame exon 20 insertion by a local test and sufficient tumor tissue available for central analysis. 2. Must have received at least 1 prior line of therapy for locally advanced or metastatic disease and no more than 2 regimens of systemic anticancer chemotherapies for locally advanced or metastatic disease. * Prior treatment with an EGFR TKI is allowed unless the participant had an objective response and subsequent progression as assessed by the investigator or treating physician during treatment with that prior TKI. Exclusion Criteria: 1. Previously received TAK-788. 2. Received small-molecule anticancer therapy (including cytotoxic chemotherapy, and investigational agents, ≤ 14 days prior to first dose of TAK-788 (except for reversible EGFR TKIs \[that is, erlotinib or gefitinib\], which are allowed in the dose escalation and expansion cohorts up to 7 days prior to the first dose of TAK-788). 3. Received antineoplastic monoclonal antibodies including immunotherapy within 28 days of the first dose of TAK-788. 4. Have been diagnosed with another primary malignancy other than NSCLC except for adequately treated non-melanoma skin cancer or cervical cancer in situ; definitively treated non-metastatic prostate cancer; or participants with another primary malignancy who are definitively relapse-free with at least 3 years elapsed since the diagnosis of the other primary malignancy. Note: This exclusion criteria does not apply to Expansion Cohort 7. 5. Received radiotherapy \<=14 days prior to the first dose of TAK-788 or has not recovered from radiotherapy-related toxicities. Palliative radiation administered outside the chest and brain, stereotactic radiosurgery (SRS), and stereotactic body radiotherapy are allowed up to 7 days prior to the first dose 6. Received a moderate or strong CYP4503A inhibitor or moderate or strong CYP3A inducer within 10 days prior to first dose of TAK-788. 7. Have undergone major surgery within 28 days prior to first dose of TAK-788. Minor surgical procedures, such as catheter placement or minimally invasive biopsy, are allowed. 8. Part 1 (dose escalation) and Expansion Cohorts 1 to 3 of Part 2 (expansion phase) only: Have symptomatic CNS metastases at screening or asymptomatic disease requiring corticosteroids to control symptoms within 7 days prior to the first dose of TAK-788. Part 3 (extension cohort) and Expansion Cohorts 4 to 7 of Part 2 (expansion phase) only: Have known active brain metastases (have either previously untreated intracranial CNS metastases or previously treated intracranial CNS metastases with radiologically documented new or progressing CNS lesions). Brain metastases are allowed if they have been treated with surgery and/or radiation and have been stable without requiring corticosteroids to control symptoms within 7 days before the first dose of TAK-788, and have no evidence of new or enlarging brain metastases. 9. Have current spinal cord compression (symptomatic or asymptomatic and detected by radiographic imaging) or leptomeningeal disease (symptomatic or asymptomatic). 10. Have significant, uncontrolled, or active cardiovascular disease. 11. Have a known history of uncontrolled hypertension. Participants with hypertension should be under treatment on study entry to control blood pressure. 12. Have prolonged QTcF interval, or being treated with medications known to be associated with the development of torsades de pointes. 13. Have an ongoing or active infection, including but not limited to, the requirement for intravenous (IV) antibiotics, or a known history of human immunodeficiency virus, hepatitis B virus (HBV), or hepatitis C virus (HCV). Testing is not required in the absence of history. 14. Currently have or have a history of interstitial lung disease, radiation pneumonitis that required steroid treatment, or drug-related pneumonitis. 15. Female participants who are lactating and breastfeeding or have a positive urine or serum pregnancy test during the screening period. Note: Female participants who are lactating will be eligible if they discontinue breastfeeding. 16. Have gastrointestinal illness or disorder that could affect oral absorption of TAK-788. 17. Have any condition or illness that, in the opinion of the investigator, might compromise participant safety or interfere with the evaluation of the safety of the drug.