Can I keep my 401K invested with my old employer and start a Simple IRA with my new employer without
You can always keep your 401(k), as long as the amount is above the legal amount that the employer is allowed to move people out.
You may want to move your 401(k) to an IRA to increase your investment options which can lead to a more balanced portfolio which may lead to a better investing experience.
You may also want to review the SIMPLE IRA options you have for costs and diversification options with a financial advisor. SIMPLE IRAs are nice because after 2 years of participation, you may be able to move your money to your own IRA without penalty. Often times I see savers gain options and reduce your costs.
All of this is best done with the help of an independent advisor creating a holistic plan.