Unless you or baby are in danger, I'd wait for things to start naturally. If the baby is head down, try and stay active - walk, dance, bounce on an exercise ball etc to gently push the baby further down and engaged. Do things that make you happy and release all the happy hormones that will start and sustain labour. Also try to stretch, massage (ie perineal massage) and open that area. You could also get a sweep done but please make sure your midwife/consultant is experienced in doing them - someone gentle.
Induction can be really hard - harder than being pregnant for a few extra days. Once they start the induction process (pessary, gels, breaking waters etc.), you can soon end up having one intervention after the other because your body may not be ready. I was induced at 40 weeks with a pessary and then the gel. The gel started contractions but no change to my cervix and no increase in dilation. They wanted to do more gel but I didn't see the point as nothing had changed and I could see it rapidly turning into them trying to break my waters after a bit more dilation, putting me on a drip to "speed things along" because once your waters break there is higher risk of infection and the clock starts ticking to have baby out; then you need an epidural because the contractions are unbearable on the drip; and finally all the drugs pumping through you can quickly lead to baby being in distress and them either having to do an instrumental delivery or emergency c section. There are other risks with interventions that you should read up about so you can make an informed decision when the time comes rather than feeling pressured by a doctor/midwife.
I went in on Saturday morning and couldn't face another round of gel on Sunday evening so asked for a break and requested to go home. I stayed until Monday morning but was a wreck after 2 nights of little sleep at the hospital (you're monitored every 4 hours) and couldn't imagine another 24hrs of interventions and prodding which the consultant pretty much told me could end with a c section given my body's slow response. Trying to look after a baby after all that is not how you want to start your journey as a mum. I asked to go home and have some rest and time to see if things would start naturally. I agreed to go to the hospital to be monitored every day to make sure baby was ok. My second baby came naturally (40+6 days) the Saturday after - my waters broke at 6pm on Friday and baby was born at 3:30am. It was very quick with just gas and air and I felt great after. I think waiting until the baby was ready to come out was the right thing.
If baby is late, there's increased risk of meconium being in your waters so put a pad on after and check for any green stuff. My first was 40+10 days and I found meconium in my waters so they induced me on the drip; I then had an epidural (the pain became unbearable), episiotomy and forceps delivery because I couldn't push because I couldn't feel anything with the epidural. It was not how I had wanted and my recovery took months but if there is any risk to the baby, you have to get baby out as quickly and safely as possible.
I hope sharing my stories helps you. Your birth story will be what it will be so try and stay calm and positive and trust your instincts. I feel like we forget sometimes but mum you know best for your baby and your body.