I'd say

Put it at the very start of your sentence. You can also drop it in the middle for a little pause. Use it when you are guessing a number or a time. For example, I'd say it takes twenty minutes. It works perfectly when someone asks for your advice. It shows you are thinking before you speak. It is much friendlier than just saying I think.

Use it when you are about to give an estimate. It is great for choosing a restaurant with friends. Use it in a meeting to suggest a new idea. It works well when you want to be polite but honest. If someone asks if their new haircut looks good, start with I'd say. It gives you a second to breathe before you answer. It is the ultimate 'polite opinion' tool.

Do not use it for facts that are 100% certain. If someone asks for the time and you are looking at a clock, don't use it. It makes you sound unsure or confused. Avoid it in very legal or high-stakes medical situations. You don't want a surgeon saying, I'd say this is the right leg. Use it for opinions, not for life-or-death data. Keep it for the subjective stuff.

English speakers, especially in the UK and US, love to avoid being too direct. Being too direct can sometimes feel aggressive or rude. I'd say is a 'hedging' phrase. It protects the speaker from being wrong. It also protects the listener's feelings. It reflects a culture that values politeness and 'saving face' in social groups. It is the linguistic equivalent of a gentle shrug.

I would say is the full, more formal version. You might hear I'd have to say when someone is making a final decision. Some people use I'd venture to say if they want to sound fancy or old-fashioned. In very casual talk, you might just hear I'd reckon. But I'd say remains the king of versatility. It fits almost everywhere.