Bevacizumab or Pemetrexed Disodium Alone or In Combination After Induction Therapy in Treating Patients With Advanced Non-Squamous Non-Small Cell Lung Cancer

Inclusion Criteria (Step 1 Induction Therapy): * Cytologically or histologically confirmed non-small cell lung cancer (NSCLC) * Predominant non-squamous histology (NSCLC not otherwise specified allowed). Mixed tumors are categorized by the predominant cell type. * Stage IV disease including M1a or M1b stages or recurrent disease * Stage IIIB (T4NX) disease with ipsilateral lung lobe allowed provided patients are not candidates for combined chemotherapy or radiotherapy * At least 12 months since prior adjuvant chemotherapy * At least 2 weeks since prior radiotherapy * Prior carboplatin allowed provided it was given as part of adjuvant chemotherapy * Patients with brain metastasis must have received local therapy to the brain and have no evidence of progression in the brain for at least 2 weeks from the time of completion of local therapy, prior to registration * ECOG (Eastern Cooperative Oncology Group) performance status 0-1 * Leukocytes ≥ 3,000/mm\^3 * Absolute neutrophil count ≥ 1,500/mm\^3 * Platelet count ≥ 100,000/mm\^3 * Total bilirubin ≤ institutional upper limit of normal (ULN) * AST and ALT ≤ 3 times ULN * Creatinine ≤ institutional ULN OR creatinine clearance ≥ 60 mL/min * Urine protein:urine dipstick ≤ 0-1+ (if \> 1+, urine protein creatinine ratio must be \< 1) * Measurable or non-measurable disease as defined by RECIST (Response Evaluation Criteria in Solid Tumours) criteria * Patients with hypertension must be adequately controlled (BP \< 150/100 mm Hg) with appropriate anti-hypertensive therapy or diet * Concurrent therapeutic anti-coagulation allowed * Fertile patients must agree to abstain from sexual intercourse or to use adequate contraceptive methods during and for at least 6 months after completion of study therapy Exclusion Criteria (Step 1 Induction Therapy): * Prior malignancy within the past 3 years except superficial melanoma, basal cell carcinoma, or carcinoma in situ * Prior systemic chemotherapy for advanced stage lung cancer * Prior use of paclitaxel, pemetrexed disodium, or bevacizumab * Major hemoptysis within the past 4 weeks * Uncontrolled intercurrent illness including, but not limited to, active infection, symptomatic congestive heart failure, unstable angina pectoris, serious cardiac arrhythmia, psychiatric illness/social situations that would limit compliance with study requirements * History of arterial thrombotic events or major bleeding within the past 12 months * Major surgery such as thoracotomy, laparotomy, craniotomy, or significant traumatic injury within 6 weeks prior to registration. Biopsy procedures and chest tube insertion are not considered major surgery for the purpose of this protocol. * Core biopsy within 7 days of registration * Significant vascular disease (e.g., aortic aneurysm requiring surgical repair or recent peripheral arterial thrombosis) within the past 6 months * Clinically significant cardiovascular disease * Abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within the past 6 months * History of serious non-healing wounds, ulcers, or bone fractures * Cavitary lesions in the lungs * Pregnant or nursing * Concurrent anti-retroviral therapy in patients with HIV infection Inclusion Criteria (Step 2 Maintenance Therapy): * Patient must have an overall stable or better response after 4 courses of induction therapy * ECOG (Eastern Cooperative Oncology Group) performance status 0-1 * Patients must be registered to Step 2 within 6 weeks of the last day of chemotherapy administration on Step 1 * Acceptable bone marrow, renal and hepatic function within 2 weeks of registration