Planning to move to Europe in the next year. Is it possible for me to keep my internet phone that I have in the US with some reasonable level of service (no dropping of conversation, no delay in conversation)?
What is the "internet phone" that you already have in the US? Your question is very ambiguous, especially for someone like you who's not a newbie.
If you have a normal phone number, like a mobile phone, which is portable to any operator within the US (most mobile phone numbers are), what you could do is port it to a SIP VoIP provider like Anveo.com, where the cost of keeping a US or Canada number at the Personal Unlimited plan is only 2 USD per month, which includes unlimited number of incoming calls for personal use. They sometime have promotions for waiving the porting fee; otherwise, it's 15 USD. There are other providers that offer a similar service, too, like VoIP.ms and CallCentric, although they are more expensive and don't support SMS, unlike Anveo.
You can buy any hardware SIP phone, and use your SIP account from above with it, and the quality of communication should be exceptional anywhere where you can have a decent broadband connection. You can also choose any of the unlocked SIP clients from the iOS and Android stores, but be aware that it's not likely to work over 2G, and some 3G and 4G operators appear to block it, too.
There are some countries--the UAE is one--where VOIP is restricted and Vonage won't work. You need an ethernet connection for Vonage. If all you have is wireless (such as in hotel rooms nowadays) you won't be able to get Vonage to work unless you have a portable router that will work off a wireless connection into which you can plug your Vonage box.