What to wear in June?

Where are you from? I just wear what I'd wear at home.

The problem with answering is that just like everywhere else, weather is difficult to predict these days. You could look at historical averages for the areas you plan to visit but averages don't tell you the whole story.
The average may be 75 somewhere but doesn't mean you won't get days in the 80s. The same for other people's experience - you can't count on it. These days I look at the actual daily weather for prior years.

Example, Frankfurt's June average is 72. In 2018 about half the days had highs in the 80s. There was also days in the mid 60s.

At this point when planning, assume that you may get some warm days as well as cool and/or rainy days. Then the day before leaving, check the 10 day forecast for the areas you plan to visit. For sure pack a rain jacket. Be prepared to layer.

Can y’all share what type of clothing would be best to wear during this time? We are a little cold natured. :-)

Sure thing. My advice is plan on light layers on top (shirt+light sweater) and pants. My wife is cold natured and normally wore a light sweater or light jacket in June/July in Munich. There was the odd warm day here or there that allowed for short sleeves or even shorts but those were exceptions and even then it was never too hot for pants.

DJ

Assume you'll get nice weather and pack accordingly. If it rains a lot or is a little cooler than expected, you can always buy an umbrella or an extra layer.

Would be meaningful to lnow where you will be (at the sea, in which cities, in mountains) and what you plan to do (walking and sightseeing, lon mountain hikes, sailing, ...). Do you plan to visit special events?

In general: This year has a more normal curve so far compared to crazy last year when we had temps around 30c on days at the end of May. With current North winds resulting from the high-pressure area over Greenland we have pleasantly cold night in Berlin around 5-10C.

I wear jeans, a blouse and have a jacket for cool days and evenings. Maybe bring a sweater, or buy a souvenir sweatshirt. I have several Frankfurt sweatshirts and fleece jackets that I wear quite a bit. :-)
It will be light out until about 22:30, so you will want to sit outside at all of the cafes, bars or restaurants.

I'll guess you will be on the go walking and hiking more than usual over the two weeks. I've had family over for years and the one thing that went wrong for a few were their shoes/boots. Get a good pair from Lowa, Vasque etc. Break them in before you travel checking for a good fit to avoid blisters and sore feet! Have good socks too...

A hat for sun protection in higher elevations. You can be sunburned in an hour or two in the mountains anytime of year. It can come in handy for rain showers too. A good sport umbrella for those windy rainy days can make a big difference in town or country.

We are going ourselves just then too, for the Berche beer festival in Erlangen, just north of Nuremberg.

https://www.berch.info/englisch.html

Depends where you are going of course - June in southern Bavaria can be cold and rainy or snow flurries. In the middle and north only a bit warmer. I have noted that our weather in Maryland is actually pretty similar to where we used to live around Nuremberg. Check Weather.com using the cities you will be visiting. For example for the next 36 hours Nuremberg will have a low of 45 and a high of 62 with showers tomorrow.

Excellent walking shoes, well broken in. I like sturdier Reebok shoes, not track shoes. Layers. Light sweater. Long sleeve shirts that can have their sleeves rolled up. T shirts or polos to wear under LS shirt or sweater. Long pants. Jeans don't dry quickly so they may not be optimal, although I wear them like everyone else in Germany. If you have lightweight down jackets they pack small, or bring a nice set of long underwear, which pack smaller. Hope you won't need them. Women probably want only one skirt if any. Plan on doing laundry every 4/5 days to give yourselves a cafe break for writing postcards and reviewing what you want to see next, and cut down on baggage. I doubt it will be warm enough to really need shorts.

An excellent website for travelers:
https://www.bahn.de/en/view/index.shtml